Inclusion Criteria:

          1. Histologically or cytologically documented adult T-cell leukemia/lymphoma (ATLL) with
             the following characteristics:

               -  Any stage of disease,

               -  Aggressive types (for definition of ATLL subtypes see Appendix H),

               -  Documented presence of ATLL cells in peripheral blood by either morphology,
                  histology, flow cytometry or gene rearrangement studies.

          2. One of the following:

               -  Received prior AZT/IFNα therapy for ≥ 2 weeks and achieved at least partial
                  hematological response defined as > 50% reduction in absolute lymphocyte count)
                  without evidence of new disease lesions or disease progression (defined as 50%
                  increase in measurable disease from nadir as in section 14.5 if imaging is
                  performed) at the time of enrollment. OR;

               -  Received chemotherapy for ≥ 2 weeks prior, followed by at least a partial
                  hematologic response ((defined as > 50% reduction in absolute lymphocyte count),
                  and without evidence of new disease lesions or disease progression (defined as
                  50% increase in absolute lymphocyte count or measurable disease from nadir as
                  specified in section 14.5 if imaging is performed) at the time of enrollment. OR;

               -  Received high-dose steroids (prednisone, methylprednisolone, or dexamethasone)
                  followed by at least a stable partial hematologic response without evidence of
                  new disease lesions or disease progression (defined as 50% increase in absolute
                  lymphocyte count or measurable disease from nadir as specified in section 14.5 if
                  imaging is performed) within 2 weeks of enrollment.

          3. Presence of residual ATLL in peripheral blood either by morphology, histology, flow
             cytometry or gene rearrangement studies (T-cell clonality) during screening prior to
             enrollment.

          4. Documented Human T-cell lymphotropic virus type 1 (HTLV-1) infection: Documentation
             may be serologic assay (ELISA) confirmed by Western blot or polymerase chain reaction
             (PCR).

          5. Measurable or evaluable disease, including presence of molecular disease as evidence
             by T-cell clonality detected by gene rearrangement studies.

          6. 18 years of age or older.

          7. Karnofsky performance status (KPS) ≥ 50% or Eastern Cooperative Oncology Group (ECOG)
             performance status ≤ 3

          8. Patients must have adequate end organ and bone marrow function as defined below:

               -  absolute neutrophil count (ANC) ≥ 1,000 cells/mm3 [Exception: Unless cytopenias
                  are secondary to ATLL]

               -  platelets (PLT) ≥ 50,000 cells/mm3 [Exception: Unless cytopenias are secondary to
                  ATLL]

               -  Adequate hepatic function:

                    -  transaminase ≤ 2.5 the institutional upper limit of normal (ULN),

                    -  total bilirubin ≤ 1.5 x the institutional upper limit of normal (ULN),
                       [Exception: Unless secondary to hepatic infiltration with lymphoma. If the
                       elevated bilirubin is felt to be secondary to Indinavir or Atazavinir
                       therapy (or anti-HIV medications), patients will be allowed to enroll.]

               -  Creatinine clearance (CrCl) ≥ 40 mL/min, [Exception: Unless secondary to renal
                  involvement by lymphoma is suspected.]

          9. Patients who are human immunodeficiency virus positive (HIV+) are also eligible.

         10. Females of childbearing potential (CBP) must have a negative serum pregnancy test
             within one week of enrollment. Women should avoid pregnancy while receiving study
             treatment. Males and females must agree to use adequate birth control during
             participation in this trial and for 3 months after completing therapy.

         11. Patients receiving erythropoietin or Granulocyte-colony stimulating factor (G-CSF)
             from baseline are eligible.

        Exclusion Criteria:

          1. Patients with progressive disease (after previous chemotherapy or AZT/IFNα) at the
             time of enrollment.

          2. Patients with chronic leukemia with favorable features, or smoldering type ATLL.

          3. Patients receiving any other investigational agents within 14 days prior to initiation
             of study therapy. (Exception: Patients actively receiving IFN-alfa-2b,
             PEG-IFN-alfa-2b, or similar forms of IFN-alfa are permitted).

          4. Uncontrolled inter-current illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure (CHF), unstable angina pectoris,
             cardiac arrhythmia, or psychiatric illness/social situations that are likely in the
             judgment of the Investigator(s) to interfere or limit compliance with study
             requirements/treatment.

          5. Pregnant or breast-feeding women.

          6. Known hypersensitivity to histone deacetylases (HDACs), zidovudine, belinostat or any
             component of the formulation(s).

          7. Acute hepatitis or decompensated liver disease unless due to lymphoma. Chronic
             hepatitis will be required to be on prophylactic treatment during the study if
             provided liver function test meet criteria listed above without evidence of cirrhosis
             to be eligible.

          8. Concurrent active malignancies, with the exception of in situ carcinoma of the cervix,
             non-metastatic, non-melanomatous skin cancer, or Kaposi's sarcoma not requiring
             systemic chemotherapy.

          9. Known New York Heart Association (NYHA) Class 3 or 4 heart disease as per Appendix D.

         10. Known ejection fraction < 45% or institutional limit of normal range

         11. Psychological, familial, sociological or geographical conditions likely in the
             judgment of the Investigator(s) to interfere or limit compliance with study
             requirements/treatment.