Ecologists are often required to estimate the number of species in a region or designated area. A number of diversity indices are available for this purpose and are based on sampling the area using quadrats or other means, and estimating the total number of species from these samples. In this paper, a novel theory and method for estimating the number of species is developed. The theory involves the use of the Laplace method for approximating asymptotic integrals. The method is shown to be successful by testing random simulated datasets. In addition, several real survey datasets are tested, including forests that contain a large number (tens to hundreds) of tree species, and an aquatic system with a large number of fish species. The method is shown to give accurate results, and in almost all cases found to be superior to existing tools for estimating diversity.