  In [DJL07] it was shown that if A is an affine hyperplane arrangement in C^n,
then at most one of the L^2-Betti numbers of its complement is non--zero. We
will prove an analogous statement for complements of any algebraic curve in
C^2. Furthermore we also recast and extend results of [LM06] in terms of
L^2-Betti numbers.
