The Solar Terrestrial Relations Observatory (STEREO) is a pair of identical satellites that will orbit the Sun so as to drift ahead of and behind Earth respectively, to give a stereo view of the Sun. STEREO is currently scheduled for launch in November 2005. One of the instrument packages that will be flown on each of the STEREO spacecrafts is the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI), which consists of an extreme ultraviolet imager, two coronagraphs, and two side-viewing heliospheric imagers to observe solar coronal mass ejections all the way from the Sun to Earth. We report here on the inner coronagraph, labeled COR1. COR1 is a classic Lyot internally occulting refractive coronagraph, adapted for the first time to be used in space. The field of view is from 1.3 to 4 solar radii. A linear polarizer is used to suppress scattered light, and to extract the polarized brightness signal from the solar corona. The optical scattering performance of the coronagraph was first modeled using both the ASAP and APART numerical modeling codes, and then tested at the Vacuum Tunnel Facility at the National Center for Atmospheric Research in Boulder, Colorado. In this report, we will focus on the COR1 optical design, the predicted optical performance, and the observed performance in the lab. We will also discuss the mechanical and thermal design, and the cleanliness requirements needed to achieve the optical performance.