ACFT TAKING OFF WRONG RWY AT AUS NEARLY COLLIDES WITH RPTR ACFT XING THAT RWY.

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WE WERE PARKED AT THE AUSTIN TERMINAL GATE. WE WERE CLRED TO AND PUSHED BACK FROM THE GATE TO THE RAMP AREA ADJACENT TO GATE. THE FO REQUESTED TAXI CLRNC FROM AUSTIN GND CTL AT APPROX XA22 LCL TIME. WE WERE CLRED BY GND CTL TO 'TAXI ALL THE WAY TO RWY 13R.' WE LEFT THE RAMP AREA ADJACENT TO GATE; TAXIED ON THE TERMINAL RAMP TO TXWY D AND FOLLOWED TXWY D TO TXWY L. WE JOINED TXWY L AND FOLLOWED IT TO TXWY H. WE THEN JOINED TXWY H (WHICH PARALLELS RWY 13R) AND TAXIED TOWARDS THE APCH END OF RWY 13R. WE COMPLETED THE TAXI CHKLIST AND I RECONFIRMED WITH THE FO THAT WE WERE CLRED ALL THE WAY TO RWY 13R (WITH NO XING RESTRS). WE BOTH AGREED THAT WE WERE CLRED WITHOUT RESTRS TO TAXI TO RWY 13R. WE CONTINUED TAXIING ON TXWY H AND AS WE APCHED RWY 17 (WHICH WE NEEDED TO CROSS TO GET TO THE APCH END OF RWY 13R) I LOOKED BOTH DIRECTIONS ON RWY 17 FOR POSSIBLE TFC (EVEN THOUGH WE WERE CLRED TO CROSS THIS RWY BECAUSE OF OUR UNRESTR TAXI CLRNC). I DID NOT SEE ANY TFC IN EITHER DIRECTION. AT APPROX XA28 LCL TIME; AS WE STARTED TO CROSS RWY 17; A CESSNA TYPE ACFT FLEW SLIGHTLY IN FRONT OF US; APPROX 25-30 FT ABOVE US AS IT WAS TAKING OFF ON RWY 17. WE CONTINUED ACROSS RWY 17 (THE CESSNA ACFT WAS QUICKLY PAST US AND CLBING). ONCE WE WERE CLR OF RWY 17; I STOPPED OUR ACFT ON TXWY H AND I ASKED GND CTL TO CONFIRM THAT WE WERE CLRED TO TAXI ALL THE WAY TO RWY 13R. HE CONFIRMED THAT WE WERE. GND CTL ALSO TOLD US (WHEN WE INQUIRED) THAT THE CESSNA ACFT HAD TAKEN OFF ON AN UNAUTH RWY. WE LATER WERE INFORMED (BY AUSTIN TWR MGMNT) THAT THE CESSNA HAD DEPARTED ON RWY 17 WHEN IT WAS ACTUALLY CLRED TO TAKE OFF ON RWY 13L.