  Computation of the holomorphic $F_1$-function describing one-loop
gravitational couplings of vectormultiplets is shown to confirm string-string
duality of the proposed dual pair consisting of the heterotic string on
$K3\times T^2$ with gauge group $U(1)^4$ and the type IIA string on the
Calabi-Yau $WP^4_{1,1,2,8,12}(24)$.
