This paper investigates the design of a 2×2 optimal space-time block code (SBTC) for a multi-input multi-output (MIMO) system using the mutual information between the finite-alphabet input and the channel output as a main design criterion over Rayleigh fading channels. Different from the STBC design methods using PEP analysis usually seen in the literature that are valid only at high SNRs, our design method can apply to any SNR region, which is of more interest from both information-theoretic and practical points of view. To accomplish this, we first develop in closed-form an expression of the mutual information. This simple criterion is helpful in finding an optimal STBC and it can be applied to any discrete constellation such as quadrature amplitude modulation (QAM) or phase shift keying (PSK). Numerical results indicate that our proposed STBCs provide a better mutual information performance than the previously wellknown STBCs, including the Damen code and Golden code.