Chemotherapy, Radiation Therapy, Rituximab, and Umbilical Cord Blood Transplant in Treating Patients With B-Cell Non-Hodgkin's Lymphoma

DISEASE CHARACTERISTICS: * Diagnosis of 1 of the following: * CD20+\* aggressive B-cell non-Hodgkin's lymphoma (NHL), including 1 of the following: * Diffuse large cell (DLC) NHL meeting 1 of the following criteria: * Relapsed disease after initial therapy but failed to mobilize or had bone marrow involvement and therefore is not suitable for an autologous stem cell transplantation * High-intermediate or high-risk, second-line, age-adjusted International Prognostic Index (IPI) score and in second complete remission (CR) or partial remission (PR) after prior autologous stem cell transplantation * Failed prior autologous stem cell transplantation and in at least PR after salvage chemotherapy * Large cell transformation of indolent NHL/chronic lymphocytic leukemia (CLL) meeting the following criteria: * CR/PR of the large cell component of disease after salvage chemotherapy or autologous stem cell transplantation * Mantle cell lymphoma meeting 1 of the following criteria: * High-risk, as defined by p53 positivity and in first CR/PR after initial therapy * Relapsed disease after initial therapy and in second or third CR/PR after salvage chemotherapy * CD20+\* indolent NHL or CLL meeting the following criteria: * Must be in second or subsequent progression (pre-allograft cytoreduction necessary but CR/PR not required) * Indolent NHL includes, but is not limited to, any of the following: * Follicular NHL * Small cell NHL * Marginal zone NHL NOTE: \*CD20 positivity must be demonstrated within the past 12 months * Relapsed disease must be biopsy proven * Prior pre-allograft cytoreduction may have included 1 of the following: * Single autologous stem cell transplantation with high-dose chemotherapy conditioning, if appropriate, and no conditioning prior to transplantation * Two or more courses of intensive combination chemotherapy (e.g., rituximab, irinotecan hydrochloride, cetuximab, epirubicin hydrochloride \[RICE\]) as appropriate according to diagnosis and prior therapy * Heavily pre-treated CLL patients in whom further combination chemotherapy is not appropriate may receive single-agent intermediate-dose cyclophosphamide for 2-3 courses * No mantle cell or DLC NHL with progressive disease at allograft work-up * No suitable matched related or unrelated donor available * Two umbilical cord blood (UCB) units available meeting the following criteria: * Units and recipient must be ≥ 4/6 HLA-A and -B antigen and DRB1 allele matched * Each unit must have ≥ 1.5 x 10\^7 total nucleated cells/recipient body weight PATIENT CHARACTERISTICS: * Karnofsky performance score 70-100% * Creatinine clearance ≥ 50 mL/min * Bilirubin \< 2.5 mg/dL * AST and ALT ≤ 3 times upper limit of normal (unless due to benign congenital hyperbilirubinemia) * Spirometry and corrected DLCO ≥ 50% normal * LVEF ≥ 40% * Albumin ≥ 2.5 g/dL * No active and uncontrolled infection at time of transplantation, including active infection with Aspergillus or other mold * No HIV positivity * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception PRIOR CONCURRENT THERAPY: * See Disease Characteristics * No more than 120 days since prior autologous stem cell transplantation * No more than 60 days since prior chemotherapy * No prior allogeneic transplantation