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 .