Inclusion Criteria:

          -  Patients with high risk multiple myeloma who are transplant candidates, in partial
             response (PR) or better; high risk will be defined as patients with any of the
             following:

               -  Fluorescence in situ hybridization showing t(4:14), t(14:16), t(14:20), gain
                  (amp) 1q; deletion (Del) 17/17p [or tp53 gene mutation/deletion by next
                  generation sequencing (NGS), or by conventional cytogenetics];

               -  Deletion 13 by conventional cytogenetic analysis;

               -  High risk signatures as determined by the GEP-70 or EMC-92 gene expression
                  profiles;

               -  Relapsed disease within 18 months of prior autologous stem cell transplant (ASCT)

          -  Patients with plasma cell leukemia who are transplant candidates

          -  Performance score of at least 70% by Karnofsky or 0 to 2 Eastern Cooperative Oncology
             Group (ECOG)

          -  Left ventricular ejection fraction greater than 40%

          -  Pulmonary function test (PFT) demonstrating a diffusion capacity of least 40%
             predicted

          -  Estimated serum creatinine clearance >= 60 ml/min (using the Cockcroft-Gault formula
             and/or serum creatinine =< 1.6 mg/dL

          -  Serum glutamate pyruvate transaminase (SGPT) less than 3 x upper limit of normal

          -  Total bilirubin less than 2 x upper limit of normal

          -  All study participants must be registered into the mandatory Revlimid Risk Evaluation
             and Mitigation Strategy (REMS) program, and be willing and able to comply with the
             requirements of the Revlimid REMS program

          -  Females of reproductive potential must adhere to the scheduled pregnancy testing as
             required in the Revlimid REMS program

          -  Men must agree to use a latex condom during sexual contact with females of child
             bearing potential even if they have had a successful vasectomy

          -  Patients must have a cord blood (CB) unit available which is matched with the patient
             at 4, 5, or 6/6 human leukocyte antigen (HLA) class I (serological) and II (molecular)
             antigens

          -  Patient or legally authorized representative able to sign informed consent

        Exclusion Criteria:

          -  Patients receiving any other investigational agents

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to melphalan

          -  Known hypersensitivity or desquamating rash to either thalidomide or lenalidomide

          -  Uncontrolled intercurrent illness including, but not limited to ongoing or active
             infection, uncontrolled hypertension (systolic > 160, diastolic > 100 despite
             antihypertensive therapy, symptomatic congestive heart failure, unstable angina
             pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would
             limit compliance with study requirements

          -  Human immunodeficiency virus (HIV)-positive patients are excluded due to increased
             risk of lethal infections when treated with myeloablative chemotherapy