Full-diversity full-rate (FDFR) space-time codes achieve both high data rate and good reliability at the cost of high decoding complexity. In this paper, we propose a low-complexity MIMO scheme that achieves both full diversity and full rate over flat fading channels for a sufficiently large number of transmit and receive antennas. The proposed scheme is constructed by applying energy spreading transforms (EST's) to multiple data streams and spatially multiplexing the streams to multiple transmit antennas. Simulation results show that the proposed FDFR scheme outperforms the threaded algebraic space-time (TAST) code, which is a FDFR code based on maximum likelihood (ML) detection, when the number of transmit antennas (with the same number of receive antennas) are three and four. However, its detection complexity is only that of a decision-feedback detector.