A very high energy γ-ray signal has been detected at the 5.5 σ level from H1426+428, an X-ray-selected BL Lacertae object at a redshift of 0.129. The object was monitored from 1995 to 1998 with the Whipple 10 m imaging atmospheric Cerenkov telescope as part of a general blazar survey; the results of these observations, although not statistically significant, were consistently positive. X-ray observations of H1426+428 during 1999 with the BeppoSAX instrument revealed that the peak of its synchrotron spectrum occurs at greater than 100 keV, leading to the prediction of observable TeV emission from this object. H1426+428 was monitored extensively at the Whipple Observatory during the 1999, 2000, and 2001 observing seasons. The strongest TeV signals were detected in 2000 and 2001. During 2001, an integral flux of 2.04 ± 0.35 × 10-11 cm-2 s-1 above 280 GeV was recorded from H1426+428. The detection of H1426+428 supports the idea that, as also seen in Mrk 501 and 1ES 2344+514, BL Lacertae objects with extremely high synchrotron peak frequencies produce γ-rays in the TeV range.