The properties of static spherically symmetric black holes, which are both electrically and magnetically charged, and which are coupled to the dilaton in the presence of a cosmological constant, , are considered. It is shown that apart from the Reissner - Nordström - de Sitter solution with constant dilaton, such solutions do not exist if (in arbitrary spacetime dimension ). However, asymptotically anti-de Sitter dyonic black-hole solutions with a non-trivial dilaton do exist if . Both these solutions and the asymptotically flat () solutions are studied numerically for arbitrary values of the dilaton coupling parameter, , in four dimensions. The asymptotically flat solutions are found to exhibit two horizons if and one horizon otherwise. For asymptotically anti-de Sitter solutions the result is similar, but the corresponding values of are altered in a non-linear fashion which depends on and the mass and charges of the black holes. All dyonic solutions with are found to have zero Hawking temperature in the extreme limit, however, regardless of the value of .