Rice's prediction come with votes being counted from the 15 October referendum in Iraq on the draft constitution. A preliminary result is not expected for days.
U.S. President George W. Bush praised the large turnout -- about 61 percent -- and the calm compared to elections of the interim government in January.
The U.S. military, however, did report that five U.S. soldiers and a Marine had been killed on 15 October in Al-Ramadi.
Reports spoke of fighting in Al-Ramadi, a rebellious Sunni Arab city west of Baghdad. Residents there said U.S. aircraft had bombed some areas on 16 October and doctors said 25 people had been killed.
The U.S. military had no immediate information on operations in the area.
For RFE/RL's full coverage of Iraq's constitutional referendum, see "Iraq Votes: Constitution Referendum"