On February 24, 2012, about 1315 mountain standard time, a Piper PA-28-161, N20GG, experienced a loss of engine power during an en route climb to cruise. The pilot subsequently made an off airport forced landing to a field near Rapid City, South Dakota. The flight instructor and a private pilot were uninjured. The airplane sustained substantial damage to the firewall. The airplane was registered to an individual and operated by Skyways LTD under the provisions of 14 Code of Federal Regulations Part 91 as an instrument instructional flight. Instrument meteorological conditions prevailed and an instrument flight rules flight plan had been filed for the flight that departed from Rapid City Regional Airport (RAP), Rapid City, South Dakota, about 1300 and was destined to Huron Regional Airport (HON), Huron, South Dakota. The flight instructor stated that during the airplane runup before takeoff from RAP, the engine speed decreased about 75 revolutions per minute (rpm) when the carburetor heat was checked. They departed in visual flight rules conditions on a heading of 020 degrees and climbed to 3,700 feet mean sea level (MSL). Upon reaching 4,500 feet MSL, they entered a "bank of clouds" and continued to climb. At 6,500 feet MSL, the airplane "seemed to quit climbing" and there was a decrease of engine speed of about 50 rpm. The flight instructor stated that full power was used during the climb to 9,000 feet MSL, the indicated airspeed was 95 knots, and the outside air temperature between 4,000 and 6,500 feet MSL was 20 degrees Fahrenheit. The flight instructor suspected carburetor ice and turned on carburetor heat. The engine speed decreased about 900 rpm and the engine began to run "very rough." The carburetor heat was left on and the airplane began to lose altitude. In order for the engine to recover, the flight instructor said that he had to lean the mixture and turn off the carburetor heat. The engine recovered for a moment and they were able to maintain altitude but not climb. The airplane was now at 4,800 feet MSL and too low for radar vectors for the ILS 32 approach at RAP. The airplane began to lose altitude during a turn towards RAP. The flight instructor turned the carburetor heat on again and the engine "reacted severely" and experienced a total loss of engine power. They descended to about 4,300 feet MSL and had a clear view of the ground at an altitude of about 800-900 feet above ground level. The flight instructor then performed a forced landing to a pasture where the airplane touched down on the incline of a hill damaging the nosewheel and firewall. The flight instructor stated that he experienced carburetor icing in other airplanes that he has flown and thought the accident airplane was no more susceptible to carburetor icing than those airplanes. He said that he had experienced carburetor icing in the accident airplane during idle and/or descent during past flights; however, he had never experienced carburetor icing in the accident airplane with climb power. The airplane was recovered by the operator and on February 26, 2012. Federal Aviation Administration (FAA) inspectors examined the airplane. The examination revealed that there was damage to the engine firewall, engine mounts, and the bulkhead behind the nose landing gear. Wrinkling of the underside skin of the airplane was also present. Fuel samples taken from the airplane did not contain contaminant. On March 5, 2012, FAA inspectors met the operator's aircraft mechanic and the flight instructor for an engine run. The airplane tail was tied down and the cowling was removed. The mechanic freed a jammed throttle linkage associated with impact damage. The flight instructor primed the engine with fuel and it was started within a few revolutions. The engine ran without any anomalies and remained running after the electric fuel pump was turned off and the fuel tank selector was selected to the opposite fuel tank. The engine ran for a few minutes at 1,000-1,300 rpm and was not run at a higher engine speed because a propeller blade was bent. The engine was shut down and the fuel strainer was removed and examined. The strainer screen was unobstructed with only a few finely sized particles. The intake hose on the carburetor heat side was removed and the cockpit carburetor heat control was pulled to the on position. The carburetor heat butterfly valve only closed about to about 75 percent of full travel. The carburetor heat control cable swivel clamp and housing (Adel Clamp) were secure and did not show evidence of movement or slippage from impact forces during the accident. Maintenance records showed that the carburetor heat butterfly valve, bushing, and control cable were worked on and adjusted during a 100-hour inspection of the airplane, which was dated September 8, 2011, at a tachometer time of 7,983.98 hours. The tachometer time at the time of the accident was 8,058.91 hours. The airplane was powered by a Lycoming O-360-A4M, serial number L-5025-36A, engine installed under supplemental type certificate (STC) SA1842NM. The engine was equipped with a Precision MA-4-4, 10-5193, serial number 75018217, carburetor. The engine time since overhaul was 2,384 hours. The STC modified Piper PA-28-151 airplanes with a Lycoming O-360-A4M engine with carburetor 10-5193. The RAP automated surface observing system recorded at 1252: temperature - 0 degrees Celsius; dew point - -4 degrees Celsius. According to FAA publication, Winter Flying Tips, P-8740-24, conditions were present for light carburetor icing at 1252. With and outside air temperature of 20 degrees Fahrenheit (-7 degrees Celsius) in 100 percent humidity, serious icing conditions at cruise of climb power were present.