The primary objectives of this randomized, double-blind, placebo-controlled, dose-escalation study are to evaluate the safety and tolerability of 3 dose levels of cidofovir when administered to renal transplant recipients with BK virus nephropathy and to identify the maximum tolerated doses (MTD) among the 3 dose levels of cidofovir in renal transplant recipients with BK virus nephropathy. The secondary study objectives are to evaluate the antiviral effect of cidofovir at each of 3 dose levels; to evaluate the pharmacokinetics (PK) of cidofovir in renal transplant recipients with underlying renal impairment; to evaluate the pharmacodynamics (PD) of cidofovir in this setting; to evaluate allograft function at the completion of the study; and to assess allograft rejection at the completion of the study. Patients with BKVN (virus nephropathy) diagnosed by positive plasma polymerase chain reaction (PCR) or renal allograft biopsy will be randomized to receive study drug within 60 days of the date of the renal biopsy or plasma PCR assay that established the diagnosis of BKVN. The study consists of three dose cohorts (0.25 mg/kg, 0.5 mg/kg and 1.0 mg/kg); each cohort will consist of approximately 12 subjects randomized 2:1 to receive either cidofovir or placebo (0.9% normal saline) to define the MTD among the three specified doses of cidofovir. Once the MTD is established, approximately 12 additional patients will be enrolled at that dose. The MTD is defined as the dose in which no more than 2 of the 8 cidofovir treated subjects experience a dose limiting toxicity (DLT). The target enrollment is 48 subjects if all dose cohorts are fully enrolled. A 25% over-enrollment may be tolerated to allow for continued enrollment of subjects in the lower dose cohort if data are under concomitant review by the Data and Safety Monitoring Board (DSMB) or to replace non-evaluable study participants. Study participants who have been randomized and have received cidofovir/placebo (in any cohort) will be considered non-evaluable if they discontinue from the study or die for any reason except toxicities definitely related to study treatment, including DLTs. These subjects may be replaced. There will be a 5-week drug administration period (4 doses) followed by a 2 week end-of-study observation and evaluation period for each cohort. At about 3 months after last dose of study infusion, a member of the research staff will assess the study participant and counsel on pregnancy status via a phone call. The study will be overseen by a DSMB who will review the data after each dose cohort is completed. The primary endpoint of the study will assess the safety and tolerability of cidofovir in kidney transplant recipients this will be assessed by enumeration of adverse events (AEs) reported by the subjects and/or investigator, and changes observed in the physical examination (including vital signs) and laboratory evaluations during the drug administration and end-of-treatment observation and evaluation periods. The severity and relationship of AEs to receipt of study drug will be determined because the primary endpoint is focusing on the safety. The secondary endpoints are the effect of cidofovir on BK virus as determined by: percentage of subjects who achieve an undetectable BK virus urine and plasma PCR between baseline and end of treatment; rate of reduction in urine and plasma BK virus load by quantitative PCR between baseline and end of treatment; and time to reduction in BK virus urine and plasma PCR; the detailed PK of cidofovir will be evaluated in subjects at the MTD; the PD of cidofovir will be assessed by quantitating the change in BK virus deoxyribonucleic aci