The problem of encoding a discrete memoryless source with respect to a single-letter fidelity criterion, using a block code of lengthnand rateR, is considered. The probability of error,p_{n}(R,D), is defined to be the minimum probability, over all such codes, that the source will generate a sequence which cannot be encoded with distortionDor less. For sufficiently largeR, thatp_{n}(R,D)decreases doubly exponentially with blocklength,nis shown. It is known thatp_{n}(R,D) = 0for some finiten, denoted byn_{0}(R,D). An upper bound ton_{0}(R,D)is also presented and numerically evaluated. The results derived hold independently of the source statistics. It is shown that a theorem of Omura and Shohara for symmetric sources is a special case of the results herein. Additionally, a useful characterization ofR \ast (D)for row-balanced distortion matrices is obtained.