In this paper, a new full-rate space-time block code (STBC) possessing a quasi-orthogonal (QO) property is proposed for QAM and 8 transmit antennas. This code is designed by serially concatenating a real constellation-rotating precoder with the Alamouti scheme. The QO property enables ML decoding to be done with joint detection of only four real symbols like the conventional minimum decoding complexity QO-STBC (MDC-QO-STBC). However, the proposed code is guaranteed to achieve full spatial diversity for general QAM unlike the MDC-QO-STBC which is specifically presented for only 4-QAM. By computer simulation results, we show that the proposed code exhibits the identical and slightly degraded error performance with the MDC-QO-STBC for 4-QAM and the Sharma's QO-STBC for 4 and 16-QAM, respectively. Finally, we present a new modified scheme of the original code so that there is no any discontinuity of transmission at each transmit antenna, without any loss of error performance.