Magnetic fields in the solar chromosphere play a key role in the energy transfer and dynamics of the solar atmosphere. Yet a direct observation of the chromospheric magnetic field remains one of the greatest challenges in solar physics. While some advances have been made for observing the Zeeman effect in strong chromospheric lines, the effect is small and difficult to detect outside sunspots. The Hanle effect offers a promising alternative; it is sensitive to weaker magnetic fields (e.g., 5-500 G for Ly-Alpha), and while its magnitude saturates at stronger magnetic fields, the linear polarization signals remain sensitive to the magnetic field orientation. The Hanle effect is not only limited to off-limb observations. Because the chromosphere is illuminated by an anisotropic radiation field, the Ly-Alpha line is predicted to show linear polarization for on-disk, near-limb regions, and magnetic field is predicted to cause a measurable depolarization. At disk center, the Ly-Alpha radiation is predicted to be negligible in the absence of magnetic field, and linearly polarized to an order of 0.3% in the presence of an inclined magnetic field. The proposed CLASP sounding rocket instrument is designed to detect 0.3% linear polarization of the Ly-Alpha line at 1.5 arcsecond spatial resolution (0.7’’ pixel size) and 10 pm spectral resolution. The instrument consists of a 30 cm aperture Cassegrain telescope and a dual-beam spectropolarimeter. The telescope employs a ``cold mirror’’ design that uses multilayer coatings to reflect only the target wavelength range into the spectropolarimeter. The polarization analyzer consists of a rotating waveplate and a polarizing beamsplitter that comprises MgF2 plates placed at Brewster’s Angle. Each output beam of the polarizing beamsplitter, representing two orthogonal linear polarizations, is dispersed and focused using a separate spherical varied-line-space grating, and imaged with a separate 512x512 CCD camera. Prototypes of key optical components have been fabricated and tested. Instrument design is being finalized, and the experiment will be proposed for a 2014 flight aboard a NASA sounding rocket.