Therapy for Patients With Untreated Age-Adjusted International Prognostic Index Low-Intermediate Risk, High-Intermediate Risk, or High Risk Diffuse Large B Cell Lymphoma

Inclusion Criteria: * Histologic diagnosis of diffuse large B cell lymphoma, PMLBL, or follicular lymphoma grade 3B confirmed by the department of hematopathology at MSKCC: Patients with discordant bone marrow involvement (i.e. involvement by small cleaved cells or small lymphocytic lymphoma) are eligible. * Tumors express CD20 as determined by immunohistochemistry. o Ki-67 evaluation of tumor tissue * Patients must have stage III, or IV disease. Patients with IIX, disease must have at least one other age-adjusted IPI risk factor. * KPS ≤ 70 * LDH \> upper limit of normal * All patients must have FDG-PET avid (minimum SUV 2.5) measurable disease * Patients must have normal baseline cardiac function based upon echocardiogram or gated blood pool scan (MUGA) with an ejection fraction ≥ 50% * Patients must have a serum creatinine of ≤ 1.5 mg/dl; if creatinine \>1.5 mg/dl creatinine clearance must be \>60 ml/minute. * Patients must have ANC\>1000/mcl and Platelets\>50,000/mcl. If patient has cytopenias due to bone marrow involvement, these requirements are not applicable. * Patients must have a bilirubin level of \< 2.0 mg/dl in the absence of a history of Gilbert's disease (or pattern consistent with Gilbert's) * Patients must be Hepatitis B surface antigen negative, Hepatitis B core antibody negative, and Hepatitis C negative. * All patients of childbearing and child creating age must be using an acceptable form of birth control from the initiation of treatment on study until 1 year after completion of chemotherapy and/or transplant. * Women who are pre-menopausal must have a negative pregnancy test * Age between 18 and 65 * Patients must be HIV negative. This test may be pending in a patient without risk factors, as determined by the patient's physician. * If patients have a history of malignancy other than cutaneous basal cell or squamous cell carcinoma, they must be disease-free for ≥ 5 years at the time of enrollment. * Patients or their guardians must be capable of providing informed consent. * Patients must be suitable to undergo stem cell transplant. Exclusion Criteria: * Any lymphoma subtype other than DLBCL, PMLBL, follicular lymphoma grade 3B * Patients with either parenchymal brain or lepto-meningeal involvement. * No more than 14 days of prednisone therapy between the diagnostic biopsy of either DLBCL, PMLBL, or follicular lymphoma grade 3B and the initiation of treatment on study. * Known pregnancy or breast-feeding * Medical illness unrelated to NHL which in the opinion of the attending physician and principal investigator will preclude administration of chemotherapy safely. This includes patients with uncontrolled infection, chronic renal insufficiency, myocardial infarction within the past 6 months, unstable angina, cardiac arrhythmias other than chronic atrial fibrillation and chronic active or persistent hepatitis, or New York Heart Association Classification III or IV heart disease. * History of any malignancy for which the disease-free interval is \<5 years, excluding curatively treated cutaneous basal cell or squamous cell carcinoma and carcinoma in-situ of the cervix.