We report on the first results from a new microwave cavity search for dark matter axions with masses above $20~mutext{eV}$. We exclude axion models with two-photon coupling $g_{agammagamma} gtrsim 2times10^{-14}~text{GeV}^{-1}$ over the range $23.55~mutext{eV} < m_a < 24.0~mutext{eV}$. These results represent two important achievements. First, we have reached cosmologically relevant sensitivity an order of magnitude higher in mass than any existing limits. Second, by incorporating a dilution refrigerator and Josephson parametric amplifier, we have demonstrated total noise approaching the standard quantum limit for the first time in an axion search.