  We consider a regular singular Sturm-Liouville operator $L:=-\frac{d^2}{dx^2}
+ \frac{q(x)}{x^2 (1-x)^2}$ on the line segment $[0,1]$. We impose certain
boundary conditions such that we obtain a semi-bounded self-adjoint operator.
It is known that the $\zeta$-function of this operator
$\zeta_L(s)=\sum_{\lambda\in\spec(L)\setminus\{0\}} \lambda^{-s}$ has a
meromorphic continuation to the whole complex plane with 0 being a regular
point. Then, according to Ray and Singer the $\zeta$-regularized determinant of
$L$ is defined by $\detz(L):=\exp(-\zeta_L'(0)).$ In this paper we are going to
express this determinant in terms of the solutions of the homogeneous
differential equation $Ly=0$ generalizing earlier work of S. Levit and U.
Smilansky, T. Dreyfus and H. Dym, and D. Burghelea, L. Friedlander and T.
Kappeler. More precisely we prove the formula $\detz(L)=\frac{\pi W(\psi,\phi)}
{2^{\nu_0+\nu_1} \Gamma(\nu_0+1)\Gamma(\nu_1+1)}.$ Here $\phi, \psi$ is a
certain fundamental system of solutions for the homogeneous equation $Ly=0$,
$W(\phi, \psi)$ denotes their Wronski determinant, and $\nu_0, \nu_1$ are
numbers related to the characteristic roots of the regular singular points $0,
1$.
