  The dual $F$-signature is a numerical invariant defined via the Frobenius
morphism in positive characteristic. It is known that the dual $F$-signature
characterizes some singularities. However, the value of the dual $F$-signature
is not known except only a few cases. In this paper, we determine the dual
$F$-signature of Cohen-Macaulay modules over two-dimensional rational double
points. The method for determining the dual $F$-signature is also valid for
determining the Hilbert-Kunz multiplicity. We discuss it in appendix.
