  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 $20$ $\mu\mathrm{eV}$. 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 $g_{a\gamma\gamma} \gtrsim 2\times10^{-14}$ $\mathrm{GeV}^{-1}$, a
factor of 2.3 above the benchmark KSVZ model, over the mass range $23.55 < m_a
< 24.0$ $\mu\mathrm{eV}$.
  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.
