There is a thin transition region (TR) in the solar atmosphere where the temperature rises from 10,000 K in the chromosphere to millions of degrees in the corona. Little is known about the mechanisms that dominate this enigmatic region other than the magnetic field plays a key role. The magnetism of the TR can only be detected by polarimetric measurements of a few ultraviolet (UV) spectral lines, the Lyα line of neutral hydrogen at 121.6 nm (the strongest line of the solar UV spectrum) being of particular interest given its sensitivity to the Hanle effect (the magnetic-field-induced modification of the scattering line polarization). We report the discovery of linear polarization produced by scattering processes in the Lyα line, obtained with the Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) rocket experiment. The Stokes profiles observed by CLASP in quiet regions of the solar disk show that the Q/I and U/I linear polarization signals are of the order of 0.1% in the line core and up to a few percent in the nearby wings, and that both have conspicuous spatial variations with scales of ˜10 arcsec. These observations help constrain theoretical models of the chromosphere-corona TR and extrapolations of the magnetic field from photospheric magnetograms. In fact, the observed spatial variation from disk to limb of polarization at the line core and wings already challenge the predictions from three-dimensional magnetohydrodynamical models of the upper solar chromosphere.