A superconducting undulator wound with Nb-Ti based conductor, will be used in the LHC as a key part of the synchrotron radiation profile monitor system. Two undulators are needed, one per each circulating beam, providing 5 T in a 60 mm bore over two periods of 280 mm each. A full scale prototype has been designed and successfully tested in the end of 2004. In this paper, the electromagnetic and the mechanical design of the undulator are summarized. The fabrication of the prototype is described and the successful cold test results, both power test and magnetic flux density measurements, are reported