The axion is a well-motivated cold dark matter (CDM) candidate first postulated to explain the absence of $CP$ violation in the strong interactions. CDM axions may be detected via their resonant conversion into photons in a haloscope detector: a tunable high-$Q$ microwave cavity maintained at cryogenic temperature, immersed a strong magnetic field, and coupled to a low-noise receiver. This dissertation reports on the design, commissioning, and first operation of the Haloscope at Yale Sensitive to Axion CDM (HAYSTAC), a new detector designed to search for CDM axions with masses above $20$ $mumathrm{eV}$. I also describe the analysis procedure developed to derive limits on axion CDM from the first HAYSTAC data run, which excluded axion models with two-photon coupling $g_{agammagamma} gtrsim 2times10^{-14}$ $mathrm{GeV}^{-1}$, a factor of 2.3 above the benchmark KSVZ model, over the mass range $23.55 < m_a < 24.0$ $mumathrm{eV}$. This result represents two important achievements. First, it demonstrates cosmologically relevant sensitivity an order of magnitude higher in mass than any existing direct limits. Second, by incorporating a dilution refrigerator and Josephson parametric amplifier, HAYSTAC has demonstrated total noise approaching the standard quantum limit for the first time in a haloscope axion search.
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