We study a portfolio selection problem in a continuous-time Markovian regime-switching model. The market in this model is, in general, incomplete. We adopt a method to complete the market based on an enlargement of the market using a set of geometric Markovian jump securities. We solve the portfolio selection problem in the enlarged market for a power utility and a logarithmic utility. Closed-form solutions for the optimal portfolio strategies and the value functions are obtained in both cases. We also establish the relationship between the optimal portfolio problems in the enlarged market and the original market.