Exact tests greatly improve the analysis of contingency tables when marginals are low. For instance, researchers often use Fisher's exact test, which is conditional, or Barnard's test, which is unconditional but needs to deal with a nuisance parameter. Here, we describe the m-test, an exact, unconditional test for the study of d x m binomial contingency tables. When comparing binomial trials, the m-test is related to Barnard's exact test. However, the nuisance parameter is integrated over all its possible values, instead of maximized or otherwise estimated. According to Monte Carlo simulations, this strategy yields a higher statistical power than other exact tests. We also provide a package to perform the m-test in R.