FLC OF A B757-200 NOTICED A SEL CESSNA TAXI ONTO THEIR RWY AND TAKE OFF WHEN THEY ALREADY HAD RECEIVED TKOF CLRNC. AS THEY ROLLED FOR TKOF; THE CESSNA WAS OBSERVED CLBING OUT ON RWY HDG. WHEN RPTR ASKED IF THE CESSNA WAS GOING TO TURN OUT; THE TWR ADVISED THAT THEY DID NOT KNOW ABOUT ANY CESSNA.

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TAXIING FOR TKOF RWY 9L AT FLL. SWITCHED TO TWR FREQ AFTER XING RWY 13/31. TAXI CHKLIST HAD ALREADY BEEN COMPLETED. WITH APPROX 1500 FT TO TAXI TO DEP END OF RWY 9L; FLL TWR CLRED US FOR TKOF BEFORE WE REQUESTED TKOF CLRNC. WE ACKNOWLEDGED THE CLRNC AND COMPLETED TKOF CHKLIST. WE THEN SAW A SINGLE ENG CESSNA TAXI RAPIDLY ACROSS THE WARM-UP AREA ONTO RWY 9L AND BEGIN TKOF ROLL. WE THOUGHT THAT WAS SOMEWHAT UNUSUAL; BUT ASSUMED THE CESSNA HAD RECEIVED TKOF CLRNC BEFORE WE SWITCHED TO TWR FREQ. NONETHELESS; WE ASKED TWR TO RECONFIRM OUR TKOF CLRNC. TWR CONFIRMED OUR CLRNC AS WE REACHED THE END OF THE RWY. WE TURNED ONTO RWY 9L AND CTL WAS XFERRED TO FO WHO BEGAN TKOF. AFTER EPR HAD BEEN SELECTED TO ENGAGE AUTOTHROTTLES; PNF (CAPT) NOTICED THAT THE CESSNA WAS AIRBORNE BUT STILL ON RWY CTRLINE. PNF XMITTED TO TWR; 'IS THE CESSNA GOING TO TURN?' AFTER A SHORT PAUSE; THE TWR'S ANSWER WAS; 'WHAT CESSNA?' IT IMMEDIATELY BECAME APPARENT THAT THE CESSNA HAD DEPARTED WITHOUT CLRNC. AS WE WERE NOW INTO OUR TKOF ROLL; IT SEEMED CLR THAT A HIGH SPD ABORT WAS NOT ADVISED. PNF RESPONDED TO TWR; 'THE CESSNA THAT JUST TOOK OFF RWY 9L; WE HAVE HIM VISUALLY AND WILL TURN L IMMEDIATELY WHEN AIRBORNE.' THE TWR THEN IDENTED THE CESSNA AND INSTRUCTED IT TO MAKE AN IMMEDIATE 20 DEG R TURN. IT COMPLIED AND LATERAL SEPARATION WAS INSTANTLY ESTABLISHED. THE CESSNA WAS CONTINUALLY KEPT IN VISUAL CONTACT BY BOTH PLTS AND COLLISION WAS NEVER IMMINENT. TWR GAVE US A NEW HDG TO FLY AND SWITCHED US TO DEP CTL. THEY ALSO SAID THEY WOULD DEAL WITH THE CESSNA PLT.