In this paper a test of fit for exponentiality based on the memoryless property is proposed. This procedure is applicable whether the scale and or the location parameters of the distribution are known or not The limiting behavior of the proposed test statistic is obtained and is shown to be normal under minimal conditions for both the null and null null distributions. The use of the proposed test is shown in an illustrative example. Also, empirical power of the test is evaluated for some commonly used alternatives using Monte Carlo methods. This shows that the test performs well compared to other procedures in the literature.