Celecoxib (Celebrex) Versus Placebo in Men With Recurrent Prostate Cancer

This study is designed as a randomized, double blind, placebo-controlled study. These are all scientific research methods used to protect the study from personal bias or prejudice. Patients who participate in this study will be randomly assigned initially to take either celecoxib (the study drug) or placebo (an inactive substance that looks like the study drug). Placebos are used to help determine if the results of a study (good or bad) result from chance or from treatment with the study drug. Patients will have a fifty-fifty chance of initial treatment with the study drug. Neither the patient nor the patient's doctor will know whether he/she is initially receiving the study drug or placebo. Participants who are assigned to initially take placebo may have the opportunity to be treated with celecoxib later in the study. If patients are found to be eligible for this study and they agree to participate, they will be randomized to either receive celecoxib or placebo to be taken by mouth (as a pill) twice a day, every day for as long as they are in the study. Patients will be given a pill diary to keep track of their medications. While in this study, they will be asked not to take any non-steroidal anti-inflammatory drugs, which includes over-the-counter ibuprofen and many other available drugs. In addition, they will be asked not to take any other selective COX-2 inhibitors (the class of drug that celecoxib is). While in this study, patients will have evaluations (physical exam and blood) to determine their response to their assigned regimen and to monitor side effects. Once a month the following will be performed: * Review of any side effects * Physical Exam * Routine blood tests (PSA level, kidney function). About one tablespoon of blood will be taken at each visit. Every other month starting with month 2: In addition to the above evaluations, on alternating months starting with month 2, additional routine blood tests (liver function, complete blood count) will be done. This will require an additional tablespoon of blood to be taken at these visits. Additional blood for research-related testing (about 2-3 tablespoons) will also be drawn during these visits. Participation in this study will be determined by how well patients tolerate the assigned regimen and by how their disease responds. Patients will continue taking their assigned regimen for 6 months as long as it is felt that their disease is stable (based on their PSA levels) and they are tolerating the treatment. After 6 months or earlier, if the disease progresses, the treatment code will be broken. If patients were on celecoxib, they will be taken off study. The patient's doctor will discuss other available treatment options with him/her at that time, including continuing celecoxib "off study". If patients were on placebo, they will then have the option to begin taking celecoxib. If patients start celecoxib treatment, they will be treated in this study with celecoxib for 6 months or until their disease progresses. After 6 months on celecoxib or sooner, if their disease progresses, they will be taken off study. The patient's doctor will discuss available treatment options with him/her at that time, including continuing celecoxib "off study". Patients who are taken "off study" will be monitored monthly for patient safety reasons throughout the duration of time that they are receiving celecoxib at the dose level used for this study (400mg twice a day).