We show how a solitonic ``magnetically'' charged $p$-brane solution of a given supergravity theory, with the magnetic charge carried by an antisymmetric tensor gauge field, can be generalized to a dyonic solution. We discuss the cases of ten-dimensional and eleven-dimensional supergravity in more detail and a new dyonic five-brane solution in ten dimensions is given. Unlike the purely electrically or magnetically charged five-brane solution the dyonic five-brane contains non-zero Ramond--Ramond fields and is therefore an intrinsically type~II solution. The solution preserves half of the type~II spacetime supersymmetries. It is obtained by applying a solution-generating $SL(2,\R) \times SL(2,\R)$ $S$~duality transformation to the purely magnetically charged five-brane solution. One of the $SL(2,\R)$ duality transformations is basically an extension to the type~II case of the six-dimensional $\Z_2$ string/string duality. We also present an action underlying the type IIB supergravity theory.

18 pages, latex file. Minor changes in text