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Lossless variable-length source coding with unequal cost function is considered for general sources. In this problem, the codeword cost instead of codeword length is important. The infimum of average codeword cost has already been determined for general sources. We consider the overflow probability of codeword cost and determine the infimum of achievable overflow threshold. Our analysis is on the basis of information-spectrum methods and hence valid through the general source.