We investigated the determination of the pole mass for the broken symmetry phase of the Abelian Higgs model in the $R_{\xi }$, Landau and the unitary gauges. In the case of the $R_{\xi }$ and Landau gauges, we obtained the same finite renormalized two-point functions, which were scale independent. Especially in the Landau gauge, the basic one-loop integral with zero mass $B(0,0)$ appearing in the renormalized two-point function which was originated from the $\xi $ dependent masses, canceled out in the resulting pole mass. In the case of the unitary gauge, the problem of renormalization occurs in the two-point function, and if we take only the finite part of the $% p^{4}$ term in the two point function, then we obtain the same result for the pole mass as in case of the $R_{\xi }$ and Landau gauges.