An unconditional test is presented for a single-sample Binomial experiment with random sample size. The test is shown to be uniformly more powerful than the standard Binomial Test, and is shown through several simulations to produce more accurate p-values as well. The primary downside of the test lies in the fact that it can be anticonservative (that is, produce too small a p-value), although it may be preferable to the ultra-conservatism of the standard test which treats the sample size as being fixed.