CABIN AND FLT CREWS OF B777-200 RESPOND TO A LAV FIRE APPARENTLY STARTED BY A SMOKING PASSENGER.

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WHEN WE WERE DSNDING THROUGH 17000 FT ON THE ARR APPROX 75 MI OUT OF ZZZ; WE GOT A CALL FROM THE FLT ATTENDANTS IN THE REAR OF THE ACFT INFORMING US THAT WE HAD A FIRE IN THE 4R LAVATORY; APPARENTLY FROM SOMEONE SMOKING IN THE LAVATORY. THE FO WAS FLYING THE AIRPLANE. THE RELIEF PLT HANDLED COMS WITH THE FLT ATTENDANTS AND WITH DISPATCH. WE IMMEDIATELY BEGAN LOOKING AT WHICH ARPTS WERE THE CLOSEST TO OUR POS. I WAS COMMUNICATING WITH ATC AND INFORMED THEM ABOUT OUR PROB. I MADE A DECISION TO DIVERT INTO ZZZ1 AND COMMUNICATED THAT TO ATC. AT THAT MOMENT; THE FLT ATTENDANTS RPTED THAT THEY HAD DISCHARGED SEVERAL WATER AND HALON FIRE EXTINGUISHERS INTO THE AREA OF THE FIRE AND THE FIRE APPEARED TO BE OUT WITH NO MORE SIGN OF SMOKE OR FLAME. SINCE THERE APPEARED TO BE NO FURTHER IMMEDIATE THREAT TO THE FLT; I CHANGED MY DECISION AND REQUESTED TO CONTINUE TO ZZZ BUT WITH A REQUEST FOR AN EXPEDITED STRAIGHT-IN APCH TO RWY 19L. ZZZ WAS LNDG ON RWYS 1R AND 1L AT THAT TIME AND WINDS WERE OUT OF THE W AT 10 KTS. ATC SET US UP FOR AN APCH TO RWY 19L. WE REQUESTED THE FIRE EQUIP TO STAND BY AND FOR LAW ENFORCEMENT TO MEET THE FLT SINCE THE FIRE WAS APPARENTLY THE RESULT OF SOMEONE SMOKING IN THE LAVATORY. AFTER AN UNEVENTFUL LNDG; WE EXITED THE RWY AND STOPPED ON THE HIGH SPD TXWY FOR AN INSPECTION BY THE FIREFIGHTERS. SINCE ALL APPEARED TO BE OK AFTER THE INSPECTION; WE ELECTED TO CONTINUE TO THE GATE AND ASKED THE FIRE CREWS TO STAND BY BEFORE ANY CARGO DOORS WERE OPENED AS A PRECAUTION IN CASE THERE WAS ANY FIRE ACTIVITY THAT COULD POSSIBLY HAVE SPREAD TO THE LOWER FUSELAGE. AFTER PARKING AT GATE; FIREFIGHTERS BOARDED THE ACFT AND CHKED OVER THE AFT LAVATORY AREA. THE PAX REMAINED ON BOARD WHILE WE WAITED FOR LAW ENFORCEMENT PERSONNEL TO COME ON BOARD SINCE THE FLT ATTENDANTS SUSPECTED THEY KNEW WHICH PAX HAD BEEN SMOKING. IT TOOK APPROX 10 MINS FOR THE POLICE TO ARRIVE AFTER WE BLOCKED IN TO THE GATE. PAX THEN BEGAN DEPLANING STARTING WITH THE FORWARD SECTION OF THE CABIN SINCE THE SUSPECTED SMOKER WAS IN THE AFT OF THE CABIN. I WOULD LIKE TO COMMEND THE ENTIRE CREW FOR HOW THEY HANDLED A STRESSFUL SITUATION IN A MINIMUM AMOUNT OF TIME. IN PARTICULAR; THE PURSER AND ANOTHER OF OUR FLT ATTENDANT CREW MEMBERS WERE THE ONES ACTIVELY FIGHTING THE FIRE; AND THE PURSER HANDLED THE CABIN DUTIES VERY PROFESSIONALLY IN THE AFTERMATH OF THE FIREFIGHTING AND LNDG. ALSO; WE HAD A HELPER PAX WHO HELPED A GREAT DEAL WITH TEARING AWAY PANELS IN THE LAVATORY WHILE THE FIREFIGHTING EFFORTS WERE PROCEEDING.