We report on the first results from a new microwave cavity search for dark matter axions with masses above 20 μ eV. We exclude axion models with two-photon coupling gaγγ ≳ 2 x 10-14 GeV-1 over the range 23.55 <m(a) <24.0 μ 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.