Finding an effective and reasonably objective method for setting priorities across scientific disciplines is something of a holy grail of science policy, and one whose urgency continues to grow. The Space Studies Board Task Group on Priorities in Space Research presented a thorough analysis of the pros and cons of priority setting and recommended that an effort to develop such a method proceed. The next phase involved this task group, and eventually the full Board, in an ambitious attempt to construct a formal, semiquantitative methodology to set priorities among major space science projects using both scientific and societal criteria. They also conducted a trial application of the methodology to a set of hypothetical space science initiatives. This is a report on the methodology and this exercise. Like a great many worthy scientific experiments, this exercise did not yield exactly what the framers had anticipated. In particular, the Board was not able to reach a consensus on the task group's methodology for setting priorities. This report contains an analysis of general issues in priority setting and presents a valuable record of the strengths and weaknesses of the task group's proposed methodology, one among many possible approaches. This report will inform continuing efforts in this area, including a recently undertaken, congressionally mandated Board study. The latter study is following a different approach toward different, but related, goals-agency organization and technology utilization are being analyzed together with the research priority-setting problem.