It is proved that any strongly prime alternative pair with a nonzero socle is either associative or a simple nonassociative alternative pair of a finite capacity. Using the results of \textit{O. Loos} on the structure of simple alternative pairs of finite capacity [Jordan pairs, Lecture Notes Math. 460 (1975; Zbl 0301.17003)], the authors then obtain the complete classification of strongly prime alternative pairs with nonzero socle.