The purpose of this study is to assess the safety and activity of a type of vaccine as immune therapy for prostate cancer. This vaccine will be made for each participant's own immune cells (called dendritic cells) obtained by blood donation. Dendritic cells are immune cells, whose role is to identify foreign antigens (bacteria, viruses, or tumor cells, for example) in the body and to activate other cells of the immune system to mount an attack on that foreign antigen. Each participant will be randomized into either Arm 1 (experimental treatment only) or Arm 2 (placebo first, then the experimental treatment). Participants will be given the vaccine and three boosters as an injection. After the placebo phase, each participant in Arm 2 will crossover to the treatment phase so that all participants will eventually receive the experimental treatment.