The performance of text-dependent speaker verification systems degrades in noisy environment and when the true speaker utters words that are not part of the verification password. Energy-based voice activity detection (VAD) algorithms cannot distinguish between the true speaker's speech and other background speech or between the speaker's verification password and other words uttered by the speaker. This paper presents a method for detecting the verification password in a text-dependent speaker verification system. Our speaker-dependent speech detection method is based on modeling the speaker in the surrounding noise. It can be used during verification, after a hidden Markov model (HMM) is trained from the speaker's enrollment data. We present some experimental results using this VAD algorithm in comparison with an energy-based VAD algorithm, and discuss the possibility of using the HMM-based VAD for rejecting faulty verification passwords of the true speaker and for rejecting impostors.