  Let Q(x,y)=0 be an hyperbola in the plane. Given real numbers $\beta
\equiv\beta^{2n)}=\{\beta_{ij}\}_{i,j\geq0,i+j\leq2n}$, with $\beta_{00}>0$,
the truncated Q-hyperbolic moment problem for \beta entails finding necessary
and sufficient conditions for the existence of a positive Borel measure \mu,
supported in Q(x,y)=0, such that $\beta_{ij}=\int y^{i}x^{j} d\mu (0\leq
i+j\leq2n)$. We prove that \beta admits a Q-representing measure \mu (as above)
if and only if the associated moment matrix $\mathcal{M}(n)(\beta)$ is positive
semidefinite, recursively generated, has a column relation Q(X,Y)=0, and the
algebraic variety $\mathcal{V}(\beta)$ associated to \beta satisfies
$card\mathcal{V}(\beta)\geq\rank\mathcal{M}(n)(\beta)$. In this case,
$rank\mathcal{M}(n)\leq2n+1$; if $rank\mathcal{M}(n)\leq2n$, then \beta admits
a $rank\mathcal{M}(n)$-atomic (minimal) Q-representing measure; if
$rank\mathcal{M}(n)=2n+1$, then \beta admits a Q-representing measure \mu
satisfying $2n+1\leqcard supp\mu\leq2n+2$.
