Inclusion Criteria:

          -  Patients must have histologically confirmed high grade serous ovarian or primary
             peritoneal or fallopian tube cancer; platinum resistant disease is defined as
             progression within 6 months after last platinum regimen

          -  Patients must have measurable disease, defined as at least one lesion that can be
             accurately measured in at least one dimension (longest diameter to be recorded for
             non-nodal lesions and short axis for nodal lesions) as >= 20 mm (>= 2 cm) with
             conventional techniques or as >= 10 mm (>= 1 cm) with spiral computed tomography (CT)
             scan, magnetic resonance imaging (MRI), or calipers by clinical exam; measurable
             disease by RECIST version (v)1.1 with at least one measurable target lesion

          -  Prior therapy: No line limit but no more than 1 prior regimens in the platinum
             resistant setting; no prior treatment targeting the ATR/checkpoint kinase 1 (CHK1)
             pathway and no prior gemcitabine as single agent; hormonal therapies immunotherapy,
             and antiangiogenic therapies (as single agents) do not count as lines; poly (adenosine
             diphosphate [ADP]-ribose) polymerases (PARP)-inhibitors count as a line of therapy
             unless given in the maintenance setting; PARP-inhibitors given as maintenance after
             platinum therapy do not count as a line of therapy; prior carboplatin/gemcitabine is
             allowed provided that there was no disease progression within 12 months after
             completion of the carboplatin/gemcitabine regimen; subjects may begin protocol
             treatment at least 4 weeks or 5 half-lives, whichever is shorter, after their last
             dose of chemotherapy or hormonal therapy, assuming they are otherwise eligible

          -  Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 (Karnofsky >= 70%)

          -  Life expectancy of greater than 6 months

          -  Leukocytes >= 3,000/mcL (within 2 weeks prior to initiation of study treatment)

          -  Absolute neutrophil count >= 1,500/mcL (within 2 weeks prior to initiation of study
             treatment)

          -  Platelets >= 100,000/mcL (within 2 weeks prior to initiation of study treatment)

          -  Total bilirubin within normal institutional limits (within 2 weeks prior to initiation
             of study treatment)

          -  Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 2.5 x institutional upper limit of normal (within 2 weeks prior to initiation of
             study treatment)

          -  Creatinine =< upper limit of institutional normal OR creatinine clearance >= 60
             mL/min/1.73 m^2 for patients with creatinine levels above institutional normal (within
             2 weeks prior to initiation of study treatment)

          -  Confirmation of availability of a formalin-fixed, paraffin-embedded (FFPE) tumor
             specimen with adequate tumor tissue (either one paraffin embedded tissue block OR 10
             5-micron unstained slides from the block on regular [non-plus] slides and 1
             hematoxylin and eosin [H&E] slide)

          -  All acute, clinically significant treatment-related toxicity from prior therapy,
             except for alopecia, must have resolved to grade =< 1 prior to study entry

          -  At least 4 weeks since major surgery or radiation therapy

          -  The effects of M6620 (VX-970) on the developing human fetus are unknown; for this
             reason and because deoxyribonucleic acid (DNA) damage inhibitors as well as other
             therapeutic agents used in this trial are known to be teratogenic, women of
             child-bearing potential must agree to use adequate contraception (hormonal or barrier
             method of birth control; abstinence) prior to study entry, for the duration of study
             participation, and 6 months after completion of study; 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

          -  No known hypersensitivity or contraindication to the components of study treatment
             (M6620 [VX-970], gemcitabine)

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

        Exclusion Criteria:

          -  Patients with primary platinum refractory disease, defined as progression while first
             line platinum based chemotherapy

          -  Patients who have had chemotherapy within 4 weeks or five half-lives, whichever is
             shorter, (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or
             those who have not recovered from adverse events due to agents administered more than
             4 weeks earlier

          -  Patients who have had radiotherapy within 4 weeks

          -  Patients who are receiving any other investigational agents

          -  Patients with known brain metastases should be excluded from this clinical trial
             because of their poor prognosis and because they often develop progressive neurologic
             dysfunction that would confound the evaluation of neurologic and other adverse events;
             a scan to confirm the absence of brain metastasis is not required

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to M6620 (VX-970) or gemcitabine

          -  M6620 (VX-970) is primarily metabolized by CYP3A4; therefore concomitant
             administration with strong inhibitors or inducers of CYP3A4 should be avoided; because
             the lists of these agents are constantly changing, it is important to regularly
             consult a frequently-updated medical reference for a list of drugs to avoid or
             minimize use of; patients receiving any medications or substances that are inhibitors
             or inducers of cytochrome P450 3A (CYP3A4 enzyme) are ineligible; as part of the
             enrollment/informed consent procedures, the patient 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 patient is considering a new over-the-counter medicine or herbal
             product

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac
             arrhythmia, or psychiatric illness/social situations that would limit compliance with
             study requirements

          -  Pregnant women are excluded from this study because M6620 (VX-970) and/or gemcitabine
             are agents with the potential for teratogenic or abortifacient effects; because there
             is an unknown but potential risk for adverse events in nursing infants secondary to
             treatment of the mother with M6620 (VX-970) and/or gemcitabine, breastfeeding should
             be discontinued if the mother is treated with M6620 (VX-970) and/or gemcitabine

          -  Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral
             therapy are ineligible because of the potential for pharmacokinetic interactions with
             M6620 (VX-970) and/or gemcitabine; in addition, these patients are at increased risk
             of lethal infections when treated with marrow-suppressive therapy; appropriate studies
             will be undertaken in patients receiving combination antiretroviral therapy when
             indicated