This paper concerns a mean variance portfolio selection problem in a complete market with unbounded random coefficients. In particular, we use adapted processes to model market coefficients, and assume that only the interest rate is bounded, while the appreciation rate, volatility and market price of risk are unbounded. Under an exponential integrability assumption of the market price of risk process, we first prove the uniqueness and existence of solutions to two backward stochastic differential equations with unbounded coefficients. Then we apply the stochastic linear quadratic control theory and the Lagrangian method to solve the problem. We represent the efficient portfolio and efficient frontier in terms of the unique solutions to the two backward stochastic differential equations. To illustrate our results, we derive explicit expressions for the efficient portfolio and efficient frontier in one example with Markovian models of a bounded interest rate and an unbounded market price of risk. (C) 2015 Elsevier Ltd. All rights reserved.