In Part A, a cohort of patients younger than 28 days (was less than 120 days; amended 6/20/95) of age receives open-label ddI monotherapy for 1 week before initiation of AZT/ddI combination therapy. After pharmacokinetic data are obtained, an additional cohort of patients receives ddI at a higher dose. An age-adjusted dose for ddI is determined for use in Part B. (NOTE: As of 2/13/95, Part A has completed accrued for infants 29 to 120 days of age.) Part B patients less than 90 days of age (was less than 180 days of age; amended 6/20/95) are randomized, on a double-blind basis, to receive ddI or AZT/ddI. All patients continue treatment until 12 months after the last patient on Part B is enrolled. PER AMENDMENT 4/8/97: Part A of this protocol is closed with accrual objectives met. Part B of the study will remain open for patient accrual until 6/2/97. Part B is designed as a 2-arm, randomized, double-blind study to assess safety, toxicity and tolerability as well as anti-HIV activity of ddI or AZT/ddI.