  We give another solution to the class number one problem by showing that
imaginary quadratic fields $\Q(\sqrt{-d})$ with class number $h(-d)=1$
correspond to integral points on a genus two curve $\mscrK_3$. In fact one can
find all rational points on $\mscrK_3$. The curve $\mscrK_3$ arises naturally
via certain coverings of curves:\ $\mscrK_3\rg\mscrK_6$,\
$\mscrK_1\rg\mscrK_2$\ with $\mscrK_2\colon y^2=2x(x^3-1)$ denoting the Heegner
curve, also in connection with the so-called Heegner-Stark covering
$\mscrK_1\rg\mscrK_s$.
