One of the biggest challenges in heliophysics is to decipher the magnetic structure of the solar chromosphere. The importance of measuring the chromospheric magnetic field is due to both the key role the chromosphere plays in energizing and structuring the outer solar atmosphere and the inability of extrapolation of photospheric fields to adequately describe this key boundary region. Over the last few years, significant progress has been made in the spectral line formation of UV lines as well as the MHD modeling of the solar atmosphere. It is found that the Hanle effect in the Lyman-alpha line (121.567 nm) is a most promising diagnostic tool for weaker magnetic fields in the chromosphere and transition region. Based on this groundbreaking research, we propose the Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) to NASA as a sounding rocket experiment, for making the first measurement of the linear polarization produced by scattering processes and the Hanle effect in the Lyman-alpha line (121.567 nm), and making the first exploration of the magnetic field in the upper chromosphere and transition region of the Sun. The CLASP instrument consists of a Cassegrain telescope, a rotating 1/2-wave plate, a dual-beam spectrograph assembly with a grating working as a beam splitter, and an identical pair of reflective polarization analyzers each equipped with a CCD camera. We propose to launch CLASP in December 2014.