The Weil Diffie-Hellman problem discovered by A. Joux (see Proc. ANTS 4, Lect. Notes in Comp. Sci., vol.1838, p.385-94, 2000) is a new primitive in cryptography. The authors propose the first provably secure three-party password-based authenticated key exchange protocol based on the Weil Diffie-Hellman problem. In the protocol, two communication entities can authenticate each other and establish a session key through the assistance of an authentication server. The proposed protocol requires only four communication steps, which is more efficient than previous solutions. The security of the proposed scheme has been proven in the random oracle model. In addition, the communication steps of the proposed protocol can be reduced to three rounds of transmissions.