To determine if intravenous (IV) and oral zidovudine (AZT) can be safely given to children aged 1 day to 3 months who were born to mothers with an HIV infection. Also to determine the correct dose of AZT for young children. Of a total of 908 pediatric AIDS cases, 78 percent have acquired HIV infection from a mother with HIV infection or at high risk for acquisition of HIV, and the number of cases in children is expected to increase over the next several years. AZT therapy may be effective in altering the course of the disease and decreasing the high mortality in these children. It is also possible that early intervention with AZT may prevent the establishment of HIV contracted before, during, or just after birth.