THE CAPT SAID HE EXPECTED SOME WX DRG THE 1ST 2 HRS OF FLT. AFTER LVLG AT FL 330, THE ACFT WAS IN CLEAR WX, BUT LGT TURBC CONTD FOR A TIME. AFTER THE TURBC HAD STOPPED & THERE WERE NO WX RTRNS ON THE RADAR, THE SEAT BELT SIGN WAS TURNED OFF, ALTHOUGH THE CAPT SUGGESTED TO THE PAX THAT THEY KEEP THEIR SEAT BELTS FASTENED. ABOUT 15 MINUTES LATER (APRX 2 HR & 6 MIN AFTER TKOF), THE ACFT ENCTRD SVR CLEAR AIR TURBC (CAT) W/O WARNING. SUBSEQUENTLY, IT PITCHED, THE AUTO-PLT DISCONNECTED & 35 OCCUPANTS WERE INJURED. THREE PAX WERE SERIOUSLY INJURED; 25 PAX & 7 FLT ATTENDANTS RCVD MINOR INJURIES. THE FLT THEN DIVERTED TO MIAMI WHERE THE INJURED WERE TREATED. A WX STUDY SHOWED THE TURBC WAS ENCTRD NEAR THE EDGE OF THE JET STREAM WHERE THE ACFT HAD CROSSED ABOUT 6000' OVER A CELL (OR GROUP OF CELLS). WINDS AT THAT POINT WERE ESTD TO BE FROM 260 DEG AT ABOUT 55 KTS. MODERATE TO SVR TURBC WAS FORECAST NEAR RAIN SHOWERS &/OR TSTMS.