We report on the first results from a new microwave cavity search for dark matter axions with masses above 20 μeV. We exclude axion models with two-photon coupling g_{aγγ}≳2×10^{-14} GeV^{-1} over the range 23.55<m_{a}<24.0 μeV. 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.