Authenticated key exchange protocols are of fundamental importance in securing communications and are now extensively deployed for use in various real-world network applications. In this work, we reveal major previously unpublished security vulnerabilities in the password-based authenticated three-party key exchange protocol according to Lee and Hwang (2010): (1) the Lee-Hwang protocol is susceptible to a man-in-the-middle attack and thus fails to achieve implicit key authentication; (2) the protocol cannot protect clients' passwords against an offline dictionary attack; and (3) the indistinguishability-based security of the protocol can be easily broken even in the presence of a passive adversary. We also propose an improved password-based authenticated three-party key exchange protocol that addresses the security vulnerabilities identified in the Lee-Hwang protocol.