GA SMA TKOF ON TXWY AT 90 DEGREE ANGLE TO ASSIGNED RWY.

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LANDED AT DBQ FOR REFUELING ON FLT FROM 3HO. TOOK ON FUEL AND WENT TO FSS FOR WX UPDATE. CALLED GND CTL ON 121.8 AND ADVISED THEM I REQUEST A PROGRESSIVE TAXI TO THE ACTIVE RWY. RWY 31 IN USE AT TIME. A LEFT TURN WAS MADE FROM FBO RAMP VIA SE TXWY TO FIRST RIGHT TURN TXWY. STOPPED ON HOLD MARK RWY 31. ADVISED BY GND CTL WE HAD APPROX 4000' OF RWY CLR FOR TKOF. WINDSOCK WAS TO OUR RIGHT--SHOWED WIND FROM APPROX 280 DEGS AT 8 KTS. WET COMPASS SHOWED 310 DEGS. DG WAS SET PREFLT AND RUN UP WAS DONE. GND CTL STATED; 'CLR FOR TKOF; YOU HAVE 4000' OF USABLE RWY IN FRONT OF YOU.' WET COMPASS SHOWED 310 DEGS ALONG WITH DG SETTING AT 310 DEGS. PWR WAS ADDED AND LIFT OFF SPD WAS ATTAINED WHEN I NOTICED WE WERE ON A TXWY. TOO LATE TO ABORT. UP ON LIFT OFF TWR STATED WE TOOK OFF FROM A TXWY. I STATED WE WERE NOT AWARE OF THIS UNTIL GND ROLL SPD WAS TOO HIGH TO ABORT AND MY WET COMPASS MUST BE INACCURATE. UPON CHKING THE COMPASS DASH MOUNT; WE FOUND IT TO BE 60-80 DEGS OUT ON NW AND SW HDGS. COMPASS WILL BE REPLACED WITH NEW ONE. ALSO; GND CTL SHOULD HAVE TAXIED US TO THE END OF RWY 31. NO ERROR COULD HAVE BEEN MADE THEN; AND COMPASS AND DG WOULD HAVE NOT MATCHED HDG.