This paper addresses the problem of the design of a precoder for multiple transmit antenna communication systems with spatially and temporally correlated fading channels. Using the theories of matrix differential calculus, the paper derives a precoder for unitary space-time codes that can exploit the spatio-temporal correlation in the time-varying fading channels. The design criterion is based on minimizing the mean square error of the channel estimates. Computer simulation results show that a significant performance gain can be achieved by using the designed precoder