A PA-31, N100FD, WAS BEING VECTORED BY DFW APPROACH CONTROL (APCH CTL) FOR A LANDING ON RWY 13L AT DALLAS-LOVE FLD. THE PLT'S LAST RECEIVED & ACKNOWLEDGED INSTRUCTIONS WERE TO DESCEND & MAINTAIN 2000' & TURN LEFT TO 190 DEG. SIMULTANEOUSLY,A CESSNA 337, N72430, DEPARTED RWY 15 AT ADDISON ARPT, UNDER TOWER CONTROL, & WAS TO MAKE A RGT CLIMBING TURN OF APPROX 180 DEG. TWO MIN AFTER TAKEOFF, THE PLT ASKED & GOT APPROVAL FOR FREQ CHNG. 24 SEC LATER, HE CONTACTED APCH CTL STATING HE HAD JUST DEPARTED ADDISON & WAS CLIMBING TO 2500' ON A 300 DEG HDG. THE CONTROLLER MISUNDERSTOOD THE PLT'S DEPARTURE POINT & INSTRUCTED HIM TO CHANGE TO FREQ 120.5. THE 2 ACFT IMMEDIATELY COLLIDED. A 2000' CEILING WAS REPORTED, 2 WITNESSES STATED THEY COULD SEE MORE BLUE SKY THAN CLDS. A WITNESS APRX 8 MI AWAY SAW A FIREBALL IN THE CLDS. INVESTIGA-TION REVEALED THE CONTROLLER, VECTORING N100FD, PREMATURELY CLEARED THE ACFT TO AN ALT WHICH WAS BELOW THE CEILING OF THE ADDISON ARPT TRAFFIC AREA & PRIOR TO REACHING THE AIRSPACE DELEGATED FOR THAT PURPOSE BY AN ATC LETTER OF AGREEMENT.