  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.
