As European leaders stepped up efforts to tackle the historic crisis, France also said it would take 24,000 more asylum-seekers under a European plan to relocate 120,000 refugees from hard-hit frontline countries.
Meanwhile, the poor and desperate kept coming, both on the land corridor through Turkey and the Balkans and on overcrowded boats in the Mediterranean on journeys that have claimed thousands of lives this year.
And migrants rescued by Italian coastguards yesterday said five of their group were still missing.
Merkel, at a joint news conference with her vice- chancellor, said: "What we are experiencing now is something that will ... Change our country in coming years."
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"We want the change to be positive, and we believe we can accomplish that."
Germany is expecting at least 10,000 more refugees to arrive today, an official in the south of the country said, after 20,000 entered over the weekend.
Germany was now seen by many abroad as a place of "hope", Merkel said, after citizens turned up in large numbers to shower the new arrivals with gifts, cash and toys.
Europe's top economy - which expects 800,000 asylum requests this year, four times last year's total - faces extra costs estimated at 10 billion euros (USD 11 billion) this year and next.
Merkel said that the federal government would contribute six billion euros for new shelters, extra police and language training in 2016.