In this paper we use fixed point techniques to obtain asymptotic stability results of the zero solution of a nonlinear neutral differential equation with variable delays. This investigation uses new conditions which allow the coefficient functions to change sign and do not require the boundedness of delays. An asymptotic stability theorem with a necessary and sufficient condition is proved. The obtained results improve and extend those due to Burton, Zhang, Raffoul, Jin and Luo, Ardjouni and Djoudi, and Djoudi and Khemis. Two examples are also given to illustrate this work.