This is a randomized (study drug assigned by chance), double-blind (neither the participant nor the physician know the study medication drug name), placebo (an inactive substance that is compared with a drug to test if the drug has a real effect in a clinical trial) controlled study to investigate safety and efficacy of administration of autologous cluster of differentiation 34 (CD34+) cells transduced with placebo or an anti-HIV-ribozyme (OZ1) in participants with HIV-1 infection. The total study duration will be 100 weeks and will include following visits: screening, pre-infusion Days 1-7, Day 8, Week 1, 4, 8, 12, 16, 20, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96 and 100. Before the administration of CD34+ final cell product, a number of procedures will be performed, including the injection of granulocyte colony-stimulating factor (G-CSF) to mobilize the CD34+ cells, apheresis and the transduction of the CD34+ cells with either OZ1 or placebo. Participants will be divided into two groups: one group will receive OZ1-containing CD34+ cells, the other group will receive CD34+ cells alone and will receive a single intravenous infusion (a fluid or a medicine delivered into a vein by way of a needle) of CD34+ cells transduced with either placebo or OZ1 gene transfer product. The final cell product contains approximately 2-20 x 10^7 cells/kilogram autologous CD34+ cell suspension transduced with either placebo or OZ1 gene transfer product. Primary efficacy will be assessed primarily by the amount of HIV ribonucleic acid (RNA) (viral load). Participants' safety will be monitored throughout the study.