We propose a new family of high-rate space-time block codes called embedded orthogonal space-time (EOS) codes. The family is parameterized by the number of transmit antennas, which can be any positive integer, and by the rate, which can be as high as half the number of transmit antennas. The proposed codes are based on a new concept called embedding, whereby information symbols of a traditional space-time code are replaced by codewords from a second space-time code. The EOS codes use orthogonal designs as this second code, which induces orthogonality in an effective channel matrix and leads to reduced-complexity decoding. The EOS codes have lower decoding complexity than previously reported space-time codes for any number of transmit antennas, and for any rate. Furthermore, simulation results show that the EOS codes outperform previous constructions for certain number of antennas and certain rates, when performance is measured by error probability in quasistatic Rayleigh fading.