PA-28 Cherokee/Archer/Dakota/Pillan/Warrior
Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Flight Instructor; Flight Crew Commercial
Flight Crew Last 90 Days 100; Flight Crew Total 1900; Flight Crew Type 1850
Workload; Situational Awareness
Aircraft Equipment Problem Critical; Inflight Event / Encounter Other / Unknown
Flight Crew Landed in Emergency Condition; General Declared Emergency
Aircraft
During a practice power off stall recovery; when the throttle was retarded from full power to cruise after the maneuver was over; the throttle cable broke; leading the engine to run at full power.  I took control from the student while bringing the [throttle] to idle to confirm we couldn't run at any less power.  The power continued to read between 2;500 and 2;700 RPM (redline for the prop).  I declared an emergency with [the TRACON]; whom I was already using for radar services; and diverted to [a nearby airport] with a longer runway than our home base.  At this point it became apparent I could not maintain level flight without overspeeding the propeller; so I allowed the aircraft to stay in a slow climb; eventually ending up near the airport at [3;500 AGL].  While on a very high downwind I contacted Tower; was cleared to land; and elected at that time to pull the mixture to idle to begin the descent to land.  I briefed the student on how I planned to land; using the mixture to add 'bursts' of power if necessary; and asked him to turn off the fuel and mags on my command.  The glide went well; I had the student secure the [engine] once I was sure we'd make the runway; and it was an uneventful touchdown; even managing to taxi clear of the runway before stopping.  Overall I feel everyone performed admirably; including my student; ATC; and airport staff upon landing.  This is a situation that could have been mishandled and end in disaster.  Further; a different student may not have been nearly as calm or helpful; which would have increased my workload dramatically.  As my student was completely collected; I was able to focus on getting us on the ground safely.  I don't have many suggestions for how this could be better; because it was about as good as it gets.
When the PA-28's throttle cable separated with the engine at full power after recovery from a training stall the instructor pilot assumed control; declared an emergency and; using excellent spur of the moment energy management techniques; landed uneventfully at a nearby airport.