low mass dark matter candidates such as the axion or hidden photon leverage potential interactions with electromagnetic fields, whereby the dark matter (of unknown mass) on rare occasion converts into a single photon. Current dark matter searches operating at microwave frequencies, use a resonant cavity to coherently accumulate the field sourced by the dark matter and use a quantum limited linear amplifier to read out the cavity signal. Here, we report the development of a novel microwave photon counting technique and use it to set a new exclusion limit on hidden photon dark matter. We constrain the kinetic mixing angle to ϵ≤1.82×10−15 in a narrow band around 6.011 GHz (24.86 μeV) with an integration time of 8.33 s. We operate a superconducting qubit to make repeated quantum non-demolition measurements of cavity photons and apply a hidden Markov model analysis to reduce the noise to 15.7 dB below the quantum limit, with performance limited by the residual population of the system. The techniques presented here will dramatically improve the sensitivity of future dark matter searches in the range of 3-30 GHz and are generally applicable to measurements that require high sensitivity to inherently low signal photon rates.