We search for the dark photon dark matter (DPDM) using a cryogenic millimeter-wave receiver. DPDM has a kinetic coupling with electromagnetic fields with a coupling constant of χ and is converted into ordinary photons at the surface of a metal plate. We search for signal of this conversion in the frequency range 18-26.5 GHz, which corresponds to the mass range 74-110 μeV/c^{2}. We observed no significant signal excess, allowing us to set an upper bound of χ<(0.3-2.0)×10^{-10} at 95% confidence level. This is the most stringent constraint to date and tighter than cosmological constraints. Improvements from previous studies are obtained by employing a cryogenic optical path and a fast spectrometer.