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 ≥ 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 Ω M = −0.2 +1.0 −0.8 if Ω Λ = 0. For a spatially flat universe composed of normal matter and a cosmological constant, we find Ω M = 0.4 +0.5 −0.4 , Ω Λ = 0.6 +0.4 −0.5. We demonstrate that with a sample of ∼ 30 objects, we should be able to determine relative luminosity distances over the range 0 < z < 0.5 with sufficient precision to measure Ω M with an uncertainty of ±0.2.