Inclusion Criteria:

          -  Signed and dated written informed consent.

          -  Male or female ≥ 18 years of age.

          -  Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1.

          -  Histologically confirmed adenocarcinoma of the small intestine that is advanced (not
             amenable to surgery) or metastatic (clinical stage IV). For the purposes of this
             study, ampullary tumors are considered a part of the duodenum and are classified as
             adenocarcinomas of the small intestine.

          -  At least one measurable lesion as defined by Response Evaluation Criteria in Solid
             Tumors (RECIST) version 1.1 criteria that has not been previously irradiated and which
             can be followed by CT or MRI.

          -  Adequate organ function including:

               -  Absolute neutrophil count (ANC) ≥ 1.5 × 109/L

               -  Platelets ≥ 100 × 109/L

               -  Hemoglobin ≥ 9/g/dL (may have been transfused)

               -  Total serum bilirubin ≤ 1.5 times upper limit of normal (ULN)

               -  Aspartate aminotransferase (AST/SGOT) and alanine aminotransferase (ALT/SGPT) ≤
                  2.5 × ULN (or ≤ 5 × ULN if liver metastases are present)

               -  Serum creatinine ≤ 1.5 x ULN or estimated creatinine clearance ≥ 30 mL/min as
                  calculated using the Cockcroft-Gault (CG) equation

          -  Archival tissue [paraffin block(s) or unstained slides from paraffin block(s)] from
             the primary tumor and/or a metastatic site judged reasonably available prior to
             initiating treatment, or willingness to undergo fresh pre-treatment tumor biopsy.
             (Prior to initiating treatment, the screening team must have documentation that an
             archival or fresh tumor specimen has been requested from a local or outside facility.
             However, physical possession of requested tissue or waiting for histological analysis
             or confirmation that an acquired specimen contains tumor tissue sufficient for
             analysis is not a requirement prior to initiating treatment.) If no archival tissue is
             available and patient consents to a fresh biopsy, but the patient's lesion is deemed
             inaccessible to safe biopsy, the patient will be allowed to enroll if otherwise
             eligible.

          -  Female patients of childbearing potential and male patients able to father children
             who have female partners of childbearing potential must agree to use one highly
             effective method (defined as less than 1% failure rate per year) and one additional
             effective method of contraception (Appendix 4) from 15 days prior to first trial
             treatment administration until at least 60 days after study participant's final dose
             of avelumab.

               -  Females of childbearing potential are defined as those who are not surgically
                  sterile or post-menopausal (i.e. patient has not had a bilateral tubal ligation,
                  a bilateral oophorectomy, or a complete hysterectomy; or has not been amenorrheic
                  for 12 months without an alternative medical cause). Post-menopausal status in
                  females under 55 years of age should be confirmed with a serum
                  follicle-stimulating hormone (FSH) level within laboratory reference range for
                  postmenopausal women.

               -  Male patients able to father children are defined as those who are not surgically
                  sterile (i.e. patient has not had a vasectomy).

          -  Serum pregnancy test (for females of childbearing potential) negative at screening.

          -  Re-enrollment of a subject that has discontinued the study as a pre-treatment screen
             failure (i.e. a consented patient who did not receive avelumab) is permitted. If
             re-enrolled, the subject must be re-consented. Only the screening procedures performed
             outside of protocol-specified timing must be repeated.

        Exclusion Criteria:

          -  There is no restriction on the number of prior therapies. However, prior therapy with
             antibody or drug specifically targeting T cell regulatory proteins, including but not
             limited to the following is not allowed: Prior immunotherapy with IL-2 or IFN-α, or an
             anti-PD-1 (including nivolumab), anti-PD-L1, anti-PD-L2, anti-CD137, or anti-cytotoxic
             T-lymphocyteassociated antigen-4 (CTLA-4) antibody (including ipilimumab), or any
             other antibody or drug specifically targeting T-cell co-stimulation or checkpoint
             pathways.

          -  Within 28 days before first dose of avelumab: Anti-cancer treatment, major surgery
             requiring general anesthesia, or the use of any investigational agent.

          -  Within 14 days before first dose of avelumab: Therapeutic or palliative radiation
             therapy. (Subjects receiving bisphosphonate or denosumab are eligible provided
             treatment was initiated at least 14 days before the first dose of avelumab.)

          -  Current use of immunosuppressive medication, except the following:

               -  Subjects are permitted the use of corticosteroids with minimal systemic
                  absorption (e.g. topical, ocular, intra-articular, intranasal, and inhaled);

               -  Systemic corticosteroids at physiologic doses ≤10 mg/day of prednisone or
                  equivalent are permitted;

               -  A brief (less than 3 weeks) course of corticosteroids for prophylaxis (e.g. CT
                  scan premedication against contrast dye allergy) or for treatment of
                  non-autoimmune conditions (e.g. delayed-type hypersensitivity reaction caused by
                  a contact allergen) is permitted.

          -  Previous malignant disease other than adenocarcinoma of the small intestine within the
             last 5 years, with the exception of basal or squamous cell carcinoma of the skin or
             cervical carcinoma in situ considered curatively treated (i.e. complete remission
             achieved at least 2 years prior to first dose of avelumab AND additional therapy not
             required while receiving study treatment).

          -  All subjects with brain metastases, expect those meeting the following criteria:

               -  Brain metastases that have been treated locally and are clinically stable for at
                  least 2 weeks prior to enrollment

               -  No ongoing neurological symptoms that are related to the brain localization of
                  the disease (sequelae that are a consequence of the treatment of the brain
                  metastases are acceptable.

               -  Subjects must be either off steroids or on a stable or decreasing dose of ≤ 10 mg
                  daily prednisone (or equivalent)

          -  Receipt of any organ transplantation including allogeneic stem-cell transplantation.

          -  Significant acute or chronic infections requiring systemic therapy.

               -  Known history of testing positive for human immunodeficiency virus (HIV), or
                  acquired immunodeficiency syndrome (AIDS).

               -  Hepatitis B virus (HBV) or Hepatitis C virus (HCV) infection at screening
                  (positive HBV surface antigen or HCV RNA if anti-HCV antibody screening test
                  positive).

          -  Active autoimmune disease with reasonable possibility of clinically significant
             deterioration when receiving an immunostimulatory agent:

               -  Subjects with Type 1 diabetes mellitus, vitiligo, psoriasis, hypo- or
                  hyperthyroid disease not requiring immunosuppressive treatment are eligible.

          -  Interstitial lung disease that is symptomatic or which may interfere with the
             detection or management of suspected drug-related pulmonary toxicity.

          -  Uncontrolled asthma [defined as having 3 or more of the following features of
             partially controlled asthma within 28 days prior to starting study treatment: Daytime
             symptoms more than twice per week, any limitation of activities, any nocturnal
             symptoms/awaking, need for reliever/rescue inhaler more than twice per week, or known
             lung function (PEF or FEV1) without administration of a bronchodilator that is < 80%
             predicted or personal best (if known)].

          -  Current symptomatic congestive heart failure (New York Heart Association > class II),
             unstable cardiac arrhythmia requiring therapy (e.g. medication or pacemaker), unstable
             angina (e.g. new, worsening or persistent chest discomfort), or uncontrolled
             hypertension (systolic > 160 mmHg or diastolic > 100mmHg). Or any of the following
             occurring within 6 months (180 days) prior to first dose of avelumab: Myocardial
             infarction, coronary/peripheral artery bypass graft, cerebrovascular accident or
             transient ischemic attack, or serious cardiac arrhythmia requiring medication. (Use of
             antihypertensive medication to control blood pressure is allowed.)

          -  Concurrent treatment with a non-permitted drug.

          -  Requirement of anticoagulant therapy with oral vitamin K antagonists such as Coumadin
             (warfarin). Low-dose anticoagulants for the maintenance of patency in a central venous
             access device or the prevention of deep vein thrombosis or pulmonary embolism is
             allowed. Therapeutic use of low molecular weight heparin is allowed.

          -  Persisting toxicity related to prior therapy that has not reduced to Grade 1 [National
             Cancer Institute Common Toxicity Criteria for Adverse Events (NCI CTCAE) version 4.03;
             however, alopecia and sensory neuropathy Grade ≤ 2 is acceptable.

          -  Known severe (Grade ≥ 3 NCI-CTCAE v4.03) hypersensitivity reactions to monoclonal
             antibodies, including hypersensitivity to the investigational agent or any component
             in its formulations, or history of anaphylaxis.

          -  Vaccination within 28 days of the first dose of avelumab and while on trial is
             prohibited, except for administration of inactivated vaccines (for example,
             inactivated influenza vaccine).

          -  Pregnant or breastfeeding females.

          -  Known alcohol or drug abuse.

          -  Prisoners or subjects who are involuntarily incarcerated.

          -  Other severe acute or chronic medical condition, including colitis, inflammatory bowl
             disease, pneumonitis, pulmonary fibrosis, or psychiatric condition including recent
             (within the past year) or active suicidal ideation or behavior; or laboratory
             abnormalities that may increase the risk associated with study participation or study
             treatment administration or may interfere with the interpretation of study results
             and, in the judgment of the investigator, would make the patient inappropriate for
             entry into this study.