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In this Letter, a bilaminate acoustic cloak designed using scattering cancellation methods is applied to the case of an acoustic sensor consisting of a hollow piezoelectric shell with mechanical absorption. The bilaminate cloak provides 20-50 dB reduction in scattering strength relative to the uncloaked configuration over the typical range of operation for an acoustic sensor, retains its ability to sensing acoustic pressure signals, and remains within the physical bounds of a passive absorber. Further, the cloak is shown to increase the range of frequencies over which there is nearly perfect phase fidelity between the acoustic signal and the voltage generated by the sensor. The feasibility of achieving the necessary fluid layer properties is demonstrated using sonic crystals with the use of readily available acoustic materials. [ABSTRACT FROM AUTHOR]