In this paper, we propose a spatially modulated space-time block coding scheme, called DT-SM, by combining the conventional double space time transmit diversity (DSTTD) with spatial modulation. First, we design a set of four basic spatial constellation (SC) codewords and an extended set of 16 SC codewords for four transmit antennas by using computer search based on the rank and determinant criterion. We then present a generalized algorithm to design SC codewords for an arbitrary number of transmit antennas greater than four. Compared with DSTTD, the proposed scheme attains the same transmit diversity order, while offering higher spectral efficiency. Besides, we derive a union bound for the bit error rate (BER) of the proposed DT-SM scheme in a quasi-static Rayleigh fading channel with spatial correlation. In order to achieve the performance of maximum-likelihood (ML) estimation at reduced detection complexity, we develop a sphere decoder based on the conventional Schnorr-Euchner sphere decoder. BER performance and detection complexity of the DT-SM scheme are evaluated by computer simulations. It is shown that the proposed DT-SM outperforms many existing MIMO and SM-based MIMO schemes at reasonable detection complexity, even in correlated MIMO channels.