The magnetic field plays a crucial role in the chromosphere and the transition region, and our poor empirical knowledge of the magnetic field in the upper chromosphere and transition region is a major impediment to advancing the understanding of the solar atmosphere. The Hanle effect promises to be a valuable alternative to Zeeman effect as a method of measuring the magnetic field in the chromosphere and transition region; it is sensitive to weaker magnetic fields, and also sensitive to tangled, unresolved field structures. CLASP is a sounding rocket experiment that aims to observe the Hanle effect polarization of the Lyman α (1215.67Å) line in the solar chromosphere and transition region, and prove the usefulness of this technique in placing constraints on the magnetic field strength and orientation in the low plasma-β region of the solar atmosphere. The Ly-α line has been chosen because it is a chromospheric/transition-region line, and because the Hanle effect polarization of this line is predicted to be sensitive to 10-250 Gauss, encompassing the range of interest. The CLASP instrument is designed to measure linear polarization in the Ly-α line with a polarization sensitivity of 0.1%. The instrument is currently funded for development. The optical design of the instrument has been finalized, and an extensive series of component-level tests are underway to validate the design.