In applied economics and science, the popularity of Laplace distribution as a probabilistic model has increased exponentially over the last decades. It had a long way from the first law for the error of measurement to one of the most used models for fitting economical and financial, biometrical and demographic, technological and engineering data. This increased interest from applied sciences motivates the Laplace distribution to be investigated in mathematical statistics and theory of probability. The nonregularity of the Laplace distribution makes known difficulties of its use in problems of testing statistical hypotheses. So, there is no the uniformly most powerful test for testing a simple hypothesis against a complex one-sided alternative for the case of Laplace distribution. In the present work, we use an asymptotic approach to solve a similar problem. We derive a test based on the sign statistic, calculate the deficiency of the test, provide approximations to the power function. Applied scientists can use the results to process their observations. For mathematicians, this book provides fertile ground for further investigations in asymptotic statistics.