The implications for magnetic monopole production of a first-order phase transition in the evolution of the early universe are discussed. Because of cosmological limits on the monopole density, strong constraints on the properties of the phase transition are required. The universe is likely to supercool, with the phase transition terminating either because the metastable state becomes unstable or because the universe is filled by ``bubbles'' of the broken symmetry state. Both scenarios are complicated by the exponential expansion of the supercooled universe, owing to the large cosmological term existing before the phase transition. To achieve the required dilution of the monopole/entropy ratio in either case would require a fortuitous arrangement of the parameters of the grand unified theory. We also comment on the implications of a first-order phase transition for the generation of a baryon excess. In the appendix, we elaborate a conjecture concerning the theory of nucleation in finite-temperature quantum field theory.After July 17, 1980, Dept. of Physics Kyoto University, Kyoto, Japan.