THE PVT-RATED STUDENT RECEIVED A WX BRIEFING BEFORE THE FLT & WAS ADVISED THAT FOG WAS FORECAST TO EXIST AT THE DESTINATION AT THE ESTIMATED TIME OF ARRIVAL. THE ESTIMATED TIME EN ROUTE WAS 3 HRS WITH 4 HRS OF FUEL ON BOARD. THE FLT DEPARTED AT 2027. THE FLT WAS CLEARED FOR AN ILS APPROACH & THE CFI LET THE STUDENT FLY PART OF THE APPROACH BUT TOOK CONTROL & PERFORMED A MISSED APPROACH AT MINIMUMS. THE FLIGHT WAS CLEARED FOR ANOTHER ILS APCH WHICH TERMINATED WITH A MISSED APCH. THE FLT THEN REQUESTED CLEARANCE TO A NEARBY ARPT. THE FLT WAS CLEARED FOR AN ILS APCH, & WHEN IT WAS INBOUND ON THE FINAL APPROACH, THE TOWER CONTROLLER WAS ADVISED THAT AN ENGINE QUIT AND THEY WERE 'OUTTA FUEL.' THE AIRPLANE WAS THEN DITCHED NEAR A BRIDGE & ALL OCCUPANTS EXITED THE AIRPLANE BEFORE IT SANK. THE AIRPLANE WAS RECOVERED 5 DAYS AFTER THE ACCIDENT & THE FUEL TANKS WERE DRAINED WHICH REVEALED THE LT TANK DID NOT CONTAIN ANY FUEL. THE RT FUEL TANK CONTAINED ABOUT 8 OUNCES OF FUEL. THE ACCIDENT OCCURRED AT 0012, APPROXIMATELY 3 HRS 45 MINUTES AFTER TAKEOFF.