THE CFI RECEIVED A FSS WEATHER BRIEFING FORECASTING SNOW SHOWERS BOTH EN ROUTE AND ON THE RETURN LEG OF THE FLIGHT. WHILE RECEIVING VECTORS DURING THE RETURN LEG, THE FLIGHT ENCOUNTERED HEAVY SNOW SHOWERS. THE DUAL STUDENT FLEW WHILE THE CFI FILED AND OBTAINED IFR CLEARANCE. THE PLANE FLEW FOR 4 MINUTES THROUGH HEAVY SNOW SHOWERS DURING THE CLEARANCE PROCEDURES. BOTH PILOTS SAID THE ENGINE QUIT. THE DUAL STUDENT ENTERED EMERGENCY DESCENT WHILE THE CFI ADDED CARBURETOR HEAT AND ATTEMPTED AN ENGINE RESTART, TO NO AVAIL. WHEN THE STUDENT SAID HE SAW THE TERRAIN, THE CFI TOOK THE FLIGHT CONTROLS AND COMPLETED AN EMERGENCY LANDING. AT TOUCHDOWN, THE NOSE WHEEL COLLAPSED IN THE PLOWED FIELD. THE PLANE SLID TO REST WITH A DAMAGED FUSELAGE AND LEFT WING. A POSTACCIDENT INSPECTION BY AN FAA INSPECTOR REVEALED AN ICE BUILDUP IN THE CARBURETOR AIR INDUCTION SYSTEM.