  Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is called
spherical provided it has a dense orbit on the flag variety $G/B$ of $G$.
Reductive spherical subgroups of simple Lie groups were classified by Kr\"amer
in 1979. In 1997, Brundan showed that each example from Kr\"amer's list also
gives rise to a spherical subgroup in the corresponding simple algebraic group
in any positive characteristic. Nevertheless, there is no classification of all
such instances in positive characteristic to date. The goal of this paper is to
complete this classification. It turns out that there is only one additional
instance (up to isogeny) in characteristic $2$ which has no counterpart in
Kr\"amer's classification.
  As one of our key tools, we prove a general deformation result for subgroup
schemes allowing us to deduce the sphericality of subgroups in positive
characteristic from this property for subgroups in characteristic $0$.
