Necessary and sufficient conditions for the zero solution of the nonlinear neutral integro-differential equationbegin{eqnarray*}&&frac{d}{dt}Big(r(t)Big[x(t)+Q(t, x(t-g_1(t)),...,x(t-g_N(t)))Big]Big)\ &&= -a(t)x(t)+ sum^{N}_{i=1}int^{t}_{t-g_i(t)}k_i(t,s)f_i(x(s))ds end{eqnarray*} to be asymptotically stable are obtained. In the process we invert the integro-differential equation and obtain an equivalent integral equation. The contraction mapping principle is used as the main mathematical tool for establishing the necessary and sufficient conditions.