Inclusion Criteria:

          -  Ability to understand and the willingness to sign a written informed consent document

          -  Participants must have histologically or cytologically confirmed cutaneous Kaposi
             sarcoma. Participants must have measurable disease with a minimum of five
             bi-dimensionally measurable KS cutaneous marker lesions. If fewer than five
             bi-dimensionally measurable marker lesions are available, the total surface area of
             the marker lesion(s) must be >= 700 mm^2

          -  Participants must have documentation of HIV status. If HIV negative, documentation of
             a negative HIV rapid test within 21 days before enrollment. If HIV positive,
             documentation of HIV-1 infection by means of any one of the following:

               -  Documentation of HIV diagnosis in the medical record by a licensed health care
                  provider

               -  Documentation of receipt of antiretroviral therapy (ART) (at least two different
                  medications that do not constitute a prescription for pre exposure prophylaxis
                  [PrEP]) by a licensed health care provider. Documentation may be a record of an
                  ART prescription in the participant's medical record, a written prescription in
                  the name of the participant for ART, or pill bottles for ART with a label showing
                  the participant's name

               -  HIV-1 ribonucleic acid (RNA) detection by a licensed HIV-1 RNA assay
                  demonstrating > 1000 RNA copies/mL

               -  Any licensed HIV screening antibody and/or HIV antibody/antigen combination assay
                  confirmed by a second licensed HIV assay such as a HIV-1 Western blot
                  confirmation or HIV rapid multispot antibody differentiation assay

                    -  Note: The term "licensed" refers to a kit that has been certified or
                       licensed by an oversight body within the participating country and validated
                       internally (e.g., United States [U.S.] Food and Drug Administration [FDA]).
                       WHO (World Health Organization) and CDC (Centers for Disease Control and
                       Prevention) guidelines mandate that confirmation of the initial test result
                       must use a test that is different from the one used for the initial
                       assessment. A reactive initial rapid test should be confirmed by either
                       another type of rapid assay or an extracellular interactome assay (E/CIA
                       )that is based on a different antigen preparation and/or different test
                       principle (e.g., indirect versus competitive), or a Western blot or a plasma
                       HIV-1 RNA viral load

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 50%)

          -  Life expectancy of greater than 3 months

          -  Absolute neutrophil count: >= 1,000/mm^3 (within 21 days before enrollment)

          -  Hemoglobin: > 8 g/dL (within 21 days before enrollment)

          -  Platelets: >= 75,000/mm^3 (within 21 days before enrollment). Platelet transfusions to
             help participants meet eligibility criteria are not allowed within 3 days before study
             enrollment

          -  Total bilirubin =< 1.5 x the upper limit of the normal range (ULN) (within 21 days
             before enrollment)

               -  If the elevated bilirubin is felt to be secondary to indinavir or atazanavir
                  therapy, then subjects will be allowed on protocol without any limit on the total
                  bilirubin if the direct bilirubin is normal

          -  Creatinine:

               -  Serum creatinine levels within normal institutional limits; or, creatinine
                  clearance >= 30 mL/min (as calculated per the Cockcroft-Gault Equation (within 21
                  days before enrollment)

          -  Participants with known history or current symptoms of cardiac disease, or history of
             treatment with cardiotoxic agents, must have a clinical risk assessment of cardiac
             function using the New York Heart Association Functional Classification within 6
             months before study enrollment. To be eligible for this trial, participants must be
             class 2B or better within 6 months before enrollment

          -  Ixazomib can cause fetal harm. For this reason and because proteasome inhibitors as
             well as other therapeutic agents used in this trial are known to be teratogenic, women
             of child-bearing potential and men must agree to use adequate contraception (barrier
             method of birth control and another method such as hormone contraception
             simultaneously; abstinence) before study entry, the duration of study participation,
             and 90 days after completion of ixazomib administration. Should a woman become
             pregnant or suspect she is pregnant while she or her partner is participating in this
             study, she should inform her treating physician immediately. Men treated or enrolled
             on this protocol must also agree to use adequate contraception or abstain from
             heterosexual contact before the study, for the duration of study participation, and
             for 90 days after completion of ixazomib administration

          -  If HIV positive, participants must have been on antiretroviral therapy with optimum
             anti-viral response for at least 6 months with no evidence of KS shrinkage in the last
             3 months. Changes of antiretroviral therapy within the prior 6 months for
             toxicity/convenience reasons are allowed (as long as participants are on a stable
             regimen for 4 weeks, but if any changes in anti-HIV therapy are due to inadequate HIV
             control and occurred within the 6 months prior to protocol consent, the patient is not
             eligible until 6 months after optimal control is reached and there is no KS regression
             in the last 3 months

          -  If HIV positive, must have been on stable anti-retroviral therapy for at least 4 weeks
             on a protease inhibitor (PI)-containing regimen or nucleoside reverse transcriptase
             inhibitor (NNRTI)-based regimen of at least three drugs. The NNRTIs efavirenz and
             etravirine are moderate CYP3A4 inducers and are allowed. There should be no intention
             to change the regimen for the duration of the study. Participants who have a high
             likelihood of better HIV management with a new antiretroviral regimen should defer
             enrollment until the changes are in place and the new cART regimen meets the 4-week
             criteria

        Exclusion Criteria:

          -  Participants who are receiving any other investigational agents

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to ixazomib or other agents used in study

          -  Chronic systemic treatment using strong CYP3A inducers (rifampin, rifapentine,
             rifabutin, carbamazepine, phenytoin, phenobarbital), or use of St. John's wort is not
             allowed. Patients who are on chronic use of strong CYP3A inducers must come off 14
             days before receiving ixazomib treatment. Because the lists of these agents are
             constantly changing, it is important to regularly consult a frequently-updated list;
             medical reference texts such as the Physicians' Desk Reference may also provide this
             information. As part of the enrollment/informed consent procedures, the participant
             will be counseled on the risk of interactions with other agents, and what to do if new
             medications need to be prescribed or if the participant is considering a new
             over-the-counter medicine or herbal product

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection; uncontrolled cardiovascular conditions, including uncontrolled
             hypertension, uncontrolled cardiac arrhythmias, symptomatic congestive heart failure,
             unstable angina, or myocardial infarction within the past 6 months; or psychiatric
             illness/social situations that would limit compliance with study requirements. This
             includes infections requiring systemic antibiotic therapy or other serious infection
             within 14 days before study enrollment but excludes ongoing antibiotic therapy for
             opportunistic infection (OI) prophylaxis

          -  Participants with a second prior or concurrent malignancy that has a natural history
             or treatment regimen that has the potential to interfere with the safety or efficacy
             assessment of the investigational regimen

          -  Pregnant women are excluded from this study because ixazomib is expected to cause
             fetal harm if used during pregnancy. It is not known if ixazomib is excreted into
             breast milk, but due to the potential for serious adverse events in a nursing infant,
             breastfeeding must be discontinued during therapy and for 90 days after the last
             ixazomib dose

          -  Participants who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for
             nitrosoureas or mitomycin C) before entering the study

          -  Participants who have not recovered from other adverse events due to prior anti-cancer
             therapy (i.e., have residual toxicity > grade 1), excluding alopecia

          -  Participants who are seropositive for hepatitis B (defined by a positive test for
             hepatitis B surface antigen [HBsAg]). All participants will be required to be screened
             for hepatitis B. Participants with resolved infection (i.e. participants who are HBsAg
             negative but positive for antibodies to hepatitis B core antigen (anti-HBc) and/or
             antibodies to hepatitis B surface antigen (anti-HBs) must be screened using real-time
             polymerase chain reaction (PCR) measurement of HBV deoxyribonucleic acid (DNA) levels.
             Participants who are PCR positive will be excluded. EXCEPTION: Participants with
             serologic findings suggestive of hepatitis B virus (HBV) vaccination (anti-HBs
             positivity as the only serologic marker) AND a known history of prior HBV vaccination,
             do not need to be tested for HBV DNA by PCR

          -  Participants diagnosed with hepatitis C who are hepatitis C antibody positive, whether
             hepatitis C RNA level is measurable or not, must have no evidence of cirrhosis and
             have liver function tests that conform to the protocol inclusion criteria

          -  Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral
             absorption or tolerance of ixazomib including difficulty swallowing

          -  Participants with grade 1 peripheral neuropathy with pain on clinical examination
             during the screening period

          -  Major surgery within 14 days before enrollment

          -  Participants with symptomatic visceral Kaposi sarcoma

          -  Participants who have had prior treatment that included a proteasome inhibitor