  On a closed Riemannian surface $(M,\bar g)$ with negative Euler
characteristic, we study the problem of finding conformal metrics with
prescribed volume $A>0$ and the property that their Gauss curvatures
$f_\lambda= f + \lambda$ are given as the sum of a prescribed function $f \in
C^\infty(M)$ and an additive constant $\lambda$. Our main tool in this study is
a new variant of the prescribed Gauss curvature flow, for which we establish
local well-posedness and global compactness results. In contrast to previous
work, our approach does not require any sign conditions on $f$. Moreover, we
exhibit conditions under which the function $f_\lambda$ is sign changing and
the standard prescribed Gauss curvature flow is not applicable.
