Deals with a continuous-time version of the Markowitz mean-variance portfolio selection problem. The proposed model is concerned with a basket of securities consisting of one bank account and multiple stocks. One distinct feature of the model is that the market parameters, including the bank interest rate and the appreciation and volatility rates of the stocks, depend on the market mode that switches between a finite number of states. It is assumed that the random regime switching is independent of the random sources that drive the stock prices. Although the model is more realistic and takes into account possible random volatility, it essentially renders the underlying market incomplete. We use a Markov-chain modulated diffusion formulation to model the problem. By using techniques of stochastic linear-quadratic control, we obtain mean-variance efficient portfolios and efficient frontiers in explicitly closed forms, based on solutions of two systems of linear ordinary differential equations. We also address related issues such as minimum variance portfolio and mutual fund theorem. The results are notably different from those for the case when there is no regime switching. Nonetheless, if the interest rate is independent of the Markov chain and is deterministic, then the results exhibit (rather unexpected) similarity to their no-regime-switching counterparts, even if the stocks appreciation and volatility rates are Markov-modulated.