Uncovering the nature of dark matter is one of the most important goals of particle physics. Light bosonic particles, such as the dark photon, are well-motivated candidates: they are generally long-lived, weakly interacting, and naturally produced in the early universe. In this work, we report on Light A^{'} Multilayer Periodic Optical SNSPD Target, a proof-of-concept experiment searching for dark photon dark matter in the eV mass range, via coherent absorption in a multilayer dielectric haloscope. Using a superconducting nanowire single-photon detector (SNSPD), we achieve efficient photon detection with a dark count rate of ∼6×10^{-6} counts/s. We find no evidence for dark photon dark matter in the mass range of ∼0.7-0.8 eV with kinetic mixing ε≳10^{-12}, improving existing limits in ε by up to a factor of 2. With future improvements to SNSPDs, our architecture could probe significant new parameter space for dark photon and axion dark matter in the meV to 10 eV mass range.