A B737-800's stabilizer trim became immobile while in level cruise at FL370 resulting in a nose down trim situation during approach and landing.

Narrative

Three hours into a four hour and thirty minute transcontinental flight; a B737-800 experienced a 'STAB OUT OF TRIM' light. Flight conditions were VMC at FL370 at a SAT of -52F. Amplifying information: We were in IMC for 40 minutes starting at the 1 hour airborne mark. The last time the trim on the aircraft worked with the autopilot off was at the two hour mark after trimming fuel. I mention this as trapped water in the trim motor or actuator was probably not the issue in my opinion. We consulted the QRH and performed the 'STAB OUT OF TRIM' procedure. Once completed and with no resolution; we performed the QRH procedure for 'STAB WILL NOT TRIM.' The stab appeared to be jammed as the manual trim handle would rotate about 3/4 of a turn; would not break the clutch with both stabilizer trim switches in cutout and actually would spin back to the original position. A nose down moment was present. We decided to descend to a lower altitude with temperatures about 10C to rule out ice as the culprit. Once level at FL210 for 10-20 minutes we attempted on numerous occasions to trim the stab via the manual trim wheel. Each time; the trim handle would rotate 3/4 and spin back to jammed position. All the while; Captain was talking to Maintenance and Dispatch. It was at this time that we learned that Maintenance worked on the stabilizer trim recently. As an added caution; we reviewed the 'JAMMED FLIGHT CONTROL' procedure and adjusted our approach accordingly. We declared an emergency; briefed the flight attendants and the passengers as to our condition and intent and set up for a long straight in approach. We descended early and configured. Nose forward forces were high at speeds below 300 KIAS as the stab was trimmed to Mach .795. Once at Flaps 10; control became easier. Once below 250 KIAS; the speed trim fail light illuminated twice and was cleared. Made a Flaps 15 approach and landing without incident. Upon post-flight; found stab to be noticeably out of the normal landing range. Contract Maintenance met us at the aircraft and we duplicated the condition for him. Even on the ground at 70F; the stab was jammed. Aircraft has a history of stabilizer trim problems. This could have been a very different outcome if the stabilizer would have jammed nose up during cruise climb to nose down to cruise descent. We got lucky and the failure was at a level attitude. Overall; our training and past military emergency experience allowed us to handle this with minimal risk. Suggest follow up with Maintenance and suggest that the aircraft be taken out of service for revenue until the real problem is discovered and corrected.