  The High-Z Supernova Search is an international collaboration to discover and
monitor type Ia supernovae (SN Ia) at $z > 0.2$ with the aim of measuring
cosmic deceleration and global curvature. Our collaboration has pursued a basic
understanding of supernovae in the nearby Universe, discovering and observing a
large sample of objects, and developing methods to measure accurate distances
with SN Ia. This paper describes the extension of this program to $z \geq 0.2$,
outlining our search techniques and follow-up program. We have devised
high-throughput filters which provide accurate two-color restframe $B$ and $V$
light curves of SN Ia, enabling us to produce precise, extinction-corrected
luminosity distances in the range $0.25 < z < 0.55$. Sources of systematic
error from K-corrections, extinction, selection effects, and evolution are
investigated, and their effects estimated. We present photometric and spectral
observations of SN 1995K, our program's first supernova, and use the data to
obtain a precise measurement of the luminosity distance to the $z=0.479$ host
galaxy. This object, when combined with a nearby sample of SN, yields an
estimate for the matter density of the Universe of $\Omega_M =
-0.2^{+1.0}_{-0.8}$ if $\Omega_\Lambda = 0$. For a spatially flat universe
composed of normal matter and a cosmological constant, we find $\Omega_M =
0.4^{+0.5}_{-0.4}$, $\Omega_\Lambda = 0.6^{+0.4}_{-0.5}$. We demonstrate that
with a sample of $\sim 30$ objects, we should be able to determine relative
luminosity distances over the range $0 < z< 0.5$ with sufficient precision to
measure $\Omega_M$ with an uncertainty of $\pm 0.2$.
