The study of the chemical steps in enzyme-catalyzed reactions represents a challenge for molecular simulation techniques. One concern is how to calculate paths for the reaction. Common techniques include the definition of a reaction coordinate in terms of a small set of (normally) geometrical variables or the determination of minimum energy paths on the potential energy surface of the reacting system. Both have disadvantages, the former because it presupposes knowledge of which variables are likely to be important for reaction and the latter because it provides a static picture and dynamical effects are ignored. In this paper, we employ the transition path sampling method developed by Chandler and co-workers, which overcomes some of these limitations. The reaction that we have chosen is the chorismate-mutase-catalyzed conversion of chorismate into prephenate, which has become something of a test case for simulation studies of enzyme mechanisms. We generated an ensemble of similar to 1000 independent transition paths for the reaction in the enzyme and another similar to 500 for the corresponding reaction in solution. A large variety of analyses of these paths was performed, but we have concentrated on characterizing the transition state ensemble, particularly the flexibility of its structures with respect to other ligands of the enzyme and the time evolution of various geometrical and energetic properties as the reaction proceeds. We have also devised an approximate technique for locating transition state structures along the paths.