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 \(\pm 0.2\).
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