Political analysts say the bases in the former Soviet bloc would raise U.S. security interests in the region at a time when Russia grows more assertive about its role on the global stage.
"Moscow, of course, sees the move as a provocation and as a long-term security threat, and will seek to extract a hefty geopolitical or strategic price for going along," said Alexandr Kliment, an analyst at political risk consultancy Eurasia Group.
Russian President Dmitry Medvedev said after meeting U.S. President George W. Bush on Monday the two sides should keep talking despite failing to overcome differences on the shield.
But Russian media quoted a Russian foreign ministry source as saying the deal complicated European security and cancelled out consultations with Moscow on the shield.
"A step has been taken . which in our view has not added to security on the European continent. More than that, it has complicated problems of security," Interfax news agency quoted a senior foreign ministry source as saying.
Rice and Schwarzenberg faced questions both about whether the system would ultimately be able to stop missiles and whether Bush's successor would follow through with the plan. Critics of the system say more tests are needed to prove it works.
"It's hard for me to believe that an American president is not going to want to have the capability to defend our territory, the territory of our allies . against that kind of missile threat," said Rice.
HURDLES
The shield still faces hurdles, including heavy opposition in the Czech Republic, a country of 10.4 million that the Soviets occupied for two decades after invading in 1968.
It also faces obstacles to ratification in the Czech parliament, where the government has just 100 seats in the 200-seat chamber. Some deputies say they will oppose it along with the Social-Democrat opposition in a vote that could come after a new U.S. administration takes over in January.
An opinion poll last month showed 68 percent of Czechs were against the shield, while only 24 percent supported it.
"We believe that this could start another arms race, and we believe this will not raise the security of the Czech Republic," said Frantisek Smrcka, who along with other protesters unfurled a huge banner shaped like a target in the Czech capital.
Rice will also travel to Bulgaria and Georgia on her European trip this week.
