Renault Magane 1.6 starts then turns off?

I have recently purhcased a Renault Megane 1.6 54 plate. I have only owned for 10 days and have had nothing but problems with mine. I am now experienceing a problem strating the car, it seems to swich on when key inserted, clutch pressed and start buttom pressed but it instantly turns off, it seems to do it more when its cold or wet. I took it to renault, who, as well as kindly charging me £93 to find NO problems, then said it could be the dephaser pulley at a cost £750!!

Can anyone please help?!?