We present a new technique for improving the sensitivity of an interferometer, phase-shift-amplified interferometry (PAI), which is based on two embedded interferometers. The internal interferometer, which is biased in anti-phase, amplifies the phase shift; the external interferometer converts this into an amplified intensity shift. PAI can improve the sensitivity of standard interferometers by an order of magnitude or more. The theory of PAI, including its enhanced immunity to relative intensity noise, phase noise, and other post-detection noise and distortion components, is presented. We experimentally demonstrate a phase-shift amplification factor of 11.