AN ACR LTT TOOK OFF WITHOUT A CURRENT WX RPT.

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I TUNED IN THE LBE ATIS AND DISCOVERED THAT THE CURRENT WX WAS BELOW OUR LNDG MINIMUMS AND HAD BEEN THAT WAY SINCE APPROX XA45 LCL. WE REQUESTED AND RECEIVED A HOLD AT THE BENJI NDB. AT APPROX XC20 LCL; THE LBE TWR ADVISED THAT THE WX HAD INCREASED TO OUR LNDG MINIMUMS AND WE THEN PERFORMED 2 UNSUCCESSFUL ILS APCHS BEFORE A THIRD ILS WAS SUCCESSFUL. FAR 135.219 STATES IN PART THAT 'NO PERSON MAY TAKE OFF AN ACFT UNDER IFR UNLESS THE LATEST WX RPTS OR FORECAST INDICATE THAT WX CONDITIONS AT THE ESTIMATED TIME OF ARR WILL BE AT OR ABOVE AUTH LNDG MINIMUMS.' OUR PIT DEP WAS NOT UNTIL XB25 LCL AND THE LBE WX WAS AND HAD BEEN BELOW OUR LNDG MINIMUMS FOR 40 MINS. I WAS PIC OF THAT FLT AND MY OPERATING POLICY IS SUCH THAT THE PF WILL CALCULATE THE WT AND BAL; RECEIVE WX INFO FROM THE COMPANY COMPUTER SYS; AND RECEIVE THE IFR CLRNC. THE PNF WILL BE RESPONSIBLE FOR LOADING AND SECURING PAX; CARGO; AND ACFT. AS I WAS LOADING PAX FOR THE FLT; THE FO DID HIS DUTIES EXCEPT; WHEN HE ATTEMPTED TO RECEIVE WX; THE COMPUTER WAS 'SIGNED OUT' OR; IN OTHER WORDS; IT WAS UNABLE TO PRODUCE THE WX RPT. I WAS UNAWARE OF THIS CIRCUMSTANCE. WE HAD RECEIVED LBE WX AT APPROX XA00 LCL AND THE CURRENT AND FORECAST THROUGHOUT THE FORECAST PERIOD WAS FOR VFR. ON THIS ASSUMPTION WE TAXIED AND DEPARTED FOR LBE. EVEN THOUGH DUTIES AND RESPONSIBILITIES MAY BE DELEGATED BY THE PIC; HE OR SHE; IN THE END; IS RESPONSIBLE FOR EVERYTHING. IN THE FUTURE I WILL BE MORE DILIGENT ABOUT EVERY ASPECT AS IT PERTAINS TO EACH FLT.