We apply a recently developed method, multicanonical algorithm, to the problem of tertiary structure prediction of peptides and proteins. As a simple example to test the effectiveness of the algorithm, Met-enkephalin is studied and the ergodicity problem, or multiple-minima problem, is shown to be overcome by this algorithm. The lowest-energy conformation obtained agrees with that determined by other efficient methods such as Monte Carlo simulated annealing. The superiority of the present method to simulated annealing lies in the fact that the relationship to the canonical ensemble remains exactly controlled. Once the multicanonical parameters are determined, only one simulation run is necessary to obtain the lowest-energy conformation and furthermore the results of this one run can be used to calculate various thermodynamic quantities at any temperature. The latter point is demonstrated by the calculation of the average potential energy and specific heat as functions of temperature.