DISEASE CHARACTERISTICS: Histologically confirmed malignancy including: - Pediatric acute lymphocytic leukemia (ALL) in first complete remission with high risk features including presence of t(4;11) or t(9;22), or extreme hyperleukocytosis (initial WBC greater than 500 K/mL), or failure to achieve a complete remission after standard induction therapy - Adult ALL or acute nonlymphoblastic leukemia (ANLL) in first complete remission with t(8;14) translocation or failure to achieve complete remission after standard induction therapy - ALL or ANLL in second or subsequent remission - Chronic myelogenous leukemia in chronic or accelerated phase - Myelodysplastic syndrome with evidence of evolution to acute myeloid leukemia Refractory anemia with excess blasts Refractory anemia with excess blasts in transformation Chronic myelomonocytic leukemia - T-lymphoblastic non-Hodgkin's lymphoma in second or subsequent remission - Stage IV neuroblastoma Must also meet all the following conditions: No HLA-ABC/DR identical related bone marrow or UCB donor No 5/6 antigen matched related bone marrow or UCB donor Condition precludes waiting to search and find a donor in the National Marrow Donor Registry Must have an available serologic matched umbilical cord blood unit in the New York Blood Center's Placental Blood Project No active CNS disease Not eligible for COBLT study (Transplantation of Banked Umbilicial Cord Blood Cells for Use in Clinical Research on Transplantation of Umbilical Cord Blood Stem and Progenitor Cells)

PATIENT CHARACTERISTICS: Age: Under 55 at time of umbilical cord blood transplantation Performance status:

Zubrod 0-1 Lansky 80-100% Karnofsky 80-100% Life expectancy: At least 3 months Hematopoietic: Adequate hematologic status at time of back-up bone marrow harvest: WBC at least 3,000/mm3 Absolute neutrophil count at least 1,000/mm3 Platelet count at least 100,000/mm3 Hepatic: ALT/AST no greater than 4 times normal Bilirubin no greater than 2.0 mg/dL Renal: Creatinine no greater than 2.0 mg/dL Creatinine clearance at least 50 mL/min Cardiovascular: Normal cardiac function by echocardiogram or radionuclide scan (shortening fraction or ejection fraction at least 80% of normal value for age) Pulmonary: FVC and FEV1 at least 60% of predicted for age For adults: DLCO at least 60% of predicted Other: No active infections at time of back-up bone marrow harvest or pretransplant reduction Not pregnant or nursing Negative pregnancy test Fertile patients must use effective contraception HIV negative

PRIOR CONCURRENT THERAPY: Biologic therapy: Not specified Chemotherapy: See Disease Characteristics Endocrine therapy: Not specified Radiotherapy: Not specified Surgery: Not specified