  We prove that if $\omega _1$ and $\omega _2$ are moderate weights and
$\mascB$ is a suitable (quasi-)Banach function space, then a necessary and
sufficient condition for the embedding $i\, :\, M (\omega _1,\mascB )\to M
(\omega _2,\mascB )$ between two modulation spaces to be compact is that the
quotient $\omega _2/\omega _1$ vanishes at infinity. Moreover we show, that the
boundedness of $\omega _2/\omega _1$ a necessary and sufficient condition for
the previous embedding to be continuous.
