  We provide a positive answer to Zariski's conjecture for families of singular
surfaces in $\mathbb C^3,$ under the condition that the family has a smooth
normalisation. As a corollary of the result, we obtain a surprising
characterization of the Whitney equisingularity of one parameter families of
$\mathcal A$ finitely determined map-germs $f_t: (\mathbb C^2,0) \to (\mathbb
C^3,0),$ in terms of the constancy of only one invariant, the Milnor number of
the double point locus.
