Hawker Horizon (Raytheon)
Air Traffic Control Fully Certified
Communication Breakdown; Confusion
ATC Issue All Types
Air Traffic Control Issued New Clearance
Human Factors; Procedure
A Hawker requested to climb to FL410 after leveling off at FL390. The RADAR Controller climbed the Hawker to FL400 and we both listened intently for the read back. The Hawker was in close proximity to a CRJ7. We were both were careful to listen to the read back which was FL400. We had shipped CRJ7 before the Hawker's request. We watched the Hawker and it appeared as if the airplane had leveled off. I watched the initial climb to FL403 and immediately notified the RADAR Controller. The Hawker was less than 5 miles when this happened. We corrected his altitude and the RADAR Controller at Sector X working CRJ7 responded to their TCAS alert. The RADAR Controller could have called the traffic before or after the request to climb was made to insure they understood that they were not getting their requested altitude right away. Also we could have waited until they passed. Or the R-Side could have re-emphasized the assigned altitude and explained when they could expect higher. For the pilot; I would recommend they listen more intently for what is said versus what they want to hear.
Enroute Controller described a loss of separation event when altitude assignment was confused.  The reporter noted if traffic information had been issued prior to the clearance the event may have been prevented.