A PIPER PA-31, N35206, (SOUTH CENTRAL AIR FLT 59) WAS CLEARED TO TAXI TO RWY 6L. THE WX AT THAT TIME WAS FOGGY WITH ABOUT 1/8 MI VISIBILITY, BUT THE RVR WAS IMPROVING & THE ARPT WAS EXPECTED SOON TO BE ACCEPTABLE FOR TAKEOFFS. THE PA-31 CREW TAXIED & WAITED IN THE HOLDING AREA FOR THE VISIBILITY TO IMPROVE. A DOUGLAS DC-10, H7339, KOREAN AIR LINES FLT 084, WAS CLEARED TO RWY 32. HOWEVER, WHILE TAXIING IN THE FOG, THE DC-10 CREW INADVERTENTLY TAXIED TO RWY 6L/24R. TOWER PSNL WERE UNABLE TO SEE THE ACFT IN THE FOG & WERE UNAWARE OF THE DC-10'S POSITION. THE DC-10 WAS CLEARED TO TAKE-OFF ON RWY 32 & THE PA-31 WAS CLEARED TO HOLD ON RWY 6L. THE DC-10 CREW TAXIED ON RWY 24R & BEGAN TAKING OFF TOWARD THE PA-31 WITH 2400 FT OF RWY REMAINING. ITS ESTIMATED TAKEOFF DISTANCE WAS 8150 FT. SUBSEQUENTLY, THE DC-10 COLLIDED WITH THE PA-31, CONTINUED OFF THE RWY & HIT STANCHION LIGHTS, SMALL TREES & ROUGH TERRAIN. AN INVESTIGATION REVEALED THE DC-10 PLT DID NOT CONFIRM HIS POSITION BY USING HIS COMPASS & THAT THERE WAS A LACK OF LEGIBLE TWY & RWY SIGNS.