A University of Illinois research center has awarded a contract valued at more than $188 million to Cray Inc. to build one of the world's most powerful supercomputers, an effort that International Business Machines Corp. worked on for several years before giving up in August.

The high-profile project—called Blue Waters—is led by the National Center for Supercomputing Applications, located at the university's Urbana-Champaign campus. Total spending on the effort, which is partly funded by the National Science Foundation, has...