Let \(f:X\to C\) be a geometrically ruled surface over a smooth projective curve of genus \(g\) over the field of complex numbers, and let \(H\) be an ample line bundle on \(X\). The authors determine precisely those pairs \((c_1,c_2)\), with \(c_1\in H^2 (X,\mathbb{Z})\) and \(c_2\in H^4 (X,\mathbb{Z})\), for which the moduli space \(M_H (c_1, c_2)\) of \(H\)-stable rank two vector bundles \(E\) with \(c_1(E) =c_1\) and \(c_2(E) =c_2\), is non-empty.