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A close-form expression of the exact pair-wise error probability (PEP) for space-time (S-T) codes in Rayleigh fading channel is derived using the probability-density function (PDF) of a sum of independent exponential distributed random variables. Since the expression requires evaluating the coefficients for partial fraction expansion, an easy analytical way is proposed for doing this. The exact PEP is subsequently used to develop a simple PEP using upper bound. Both PEPs are used for error rate evaluation. Numerical calculations and Monte Carlo computer simulation have been used to study the accuracies of these PEPs for error rate evaluation of a rotation-based diagonal S-T code (D code) in Rayleigh fading channels. Four other PEPs based on different bounds are also studied for comparison. Results show that our derived close-form PEP is an exact PEP and our proposed PEP is a very tight bound to the exact PEP.