High sampling rates are required to fully characterize some acoustical systems, but capturing the system's high-frequency roll-off decreases the signal-to-noise ratio (SNR). Band-pass filtering can improve the SNR, but may create an undesirable pre-response. An iterative procedure is developed to measure impulse responses (IRs) with an improved SNR and a constrained pre-response. First, a quick measurement provides information about the system and ambient noise. A second, longer measurement is then performed, and a suitable band-pass filter is applied to the recorded signal. Experimental results show that the proposed procedure achieves an SNR of 37 dB with a peak pre-response amplitude of <0.2% of the IR peak, whereas a conventional technique achieves an SNR of 32 dB with a peak pre-response amplitude of 16%.