What was found
AFTER LEVELING OFF AT FL400, MY FIRST OFFICER MENTIONED TO ME THAT THE CABIN RATE OF CLIMB WAS STILL INDICATING A 1000 FPM INCREASE AND SHE WAS WATCHING IT CLOSELY. THE CABIN ALTITUDE THEN CLIMBED HIGHER THAN USUAL, TO 8000 FT. F/O PULLED OUT THE QRH IN ANTICIPATION OF A CABIN ALTITUDE ALERT. AS SHE PULLED THE QRH, WE DID INDEED, GET AN AMBER CABIN ALTITUDE ALERT AND THE CABIN CONTINUED TO CLIMB. WE ASKED FOR A LOWER ALTITUDE AND VECTORS TO RDU AND DECIDED TO ERR ON THE SIDE OF CAUTION BY DECLARING AN EMERGENCY. WE WERE ON TOP OF A VERY STRONG LINE OF STORMS AND OPTIONS WERE LIMITED ON WHERE TO GO. AFTER DECLARING THE EMERGENCY, INFORMED THE PAX THAT THE MASKS MAY COME DOWN DUE TO A PRESSURIZATION ISSUE AND TO PUT THEM ON IF THE DID, AND WE WERE DESCENDING TO A LOWER ALTITUDE AND MAKING A PRECAUTIONARY LANDING. WHEN WE GOT LOWER, THE CABIN ALTITUDE LEVELED OFF AND THAT PREVENTED THE CABIN MASKS FROM COMING DOWN. WE CONTINUED ON TO LAND SAFELY AHEAD OF THE STORM LINE AT RDU.