  In this paper, we find an upper bound for the exponent of the Schur
multiplier of a pair $(G,N)$ of finite $p$-groups, when $N$ admits a complement
in $G$. As a consequence, we show that the exponent of the Schur multiplier of
a pair $(G,N)$ divides $\exp(N)$ if $(G,N)$ is a pair of finite $p$-groups of
class at most $p-1$. We also prove that if $N$ is powerfully embedded in $G$,
then the exponent of the Schur multiplier of a pair $(G,N)$ divides $\exp(N)$.
