A generalized Exact Test for n x m contingency tables based upon methodology of the Fisher- Yates Exact Test for 2 x 2 tables and the chi-square statistic is presented. The General Exact Test is a multi-tailed statistic with no minimum expected matrix cell values required. A computer program for the General Exact Test is described and presented in the computer language BASIC. The number of possible tables that need to be considered for an Exact Test can be very large. However, when expected cell sizes are large the chi-square distribution can be used for an approximation to the probability. The generalized Exact Test should be very useful for n x m contingency tables where categories are nominally measured and some expected values are small (e.g. taxonomic problems involving type specimens, or protein electrophoretic data), since other available statistical tests require arbitrary lumping of categories and/or populations.