By Laurence Peter
Kosovo Albanians protested against the EU mission on Tuesday
The EU's police and justice mission will start deploying throughout Kosovo on 9 December, including to Serb-controlled areas, EU officials say.
Under the plan, approved by the UN Security Council, nearly 2,000 EU police, justice and customs officials will take over UN duties in Kosovo.
But the UN mission - called Unmik - will retain a political role.
The EU's "Eulex" mission was supposed to start earlier. Kosovo declared independence from Serbia in February.
Several thousand Kosovo Albanians demonstrated in the capital Pristina against the Eulex deployment on Tuesday - the day it was originally scheduled to start.
Protesters held banners saying "Kosovo in the EU, not under the EU" and "Eulex made in Serbia".
The protest organiser and leader of the Self-determination Movement, Albin Kurti, voiced fears that the EU officials would "have immunity from the law - they will also be above the law".
Eulex police spokeswoman Karin Limdal said Eulex would operate "all over Kosovo - in the Serb-controlled areas too".
About 1,300 Eulex police, judges, prosecutors, customs officers and prison officers will start deploying on 9 December, rising to about 1,900 in a few months' time, she told the BBC News website.
Eulex says it is now "ready to go operational" in Kosovo
Meanwhile, Serbia and Russia remain strongly opposed to Kosovo's independence declaration, refusing to recognise the territory.
So far 52 countries have recognised Kosovo, including the United States and many EU member states. More than 100 have not recognised it.
Serbia had objected to Eulex having any role in Kosovo until the plan was altered to keep the UN in Serb-majority areas.
But Eulex understands that Belgrade will not oppose an EU presence in Serb-majority areas, which are mostly in the north. Kosovo's population is 90% Albanian, with a minority of Serbs.
An Unmik spokesman, Alexander Ivanko, confirmed on Tuesday that the UN mission would be downsized in the Serb-majority areas, as well as the rest of Kosovo.
"As soon as Eulex deploys throughout Kosovo we'll curtail our operations in the field of law and justice," Mr Ivanko told the BBC News website.
The UN says Eulex will operate under a UN "umbrella" headed by the UN secretary-general's new special representative, Lamberto Zannier.
Mr Ivanko said Unmik "will no longer administer Kosovo once Eulex deploys - we will only deal with political monitoring and reporting, as mandated".
The deployment plan, approved by the UN Security Council on 27 November after difficult negotiations, does not explicitly refer to the thorny issue of Kosovo independence.
Unmik has been in charge in Kosovo since 1999 - the year that a Nato bombing campaign triggered a withdrawal of Serb forces, who had been widely condemned for human rights abuses.
Nearly 16,000 Nato-led K-For troops remain in Kosovo, maintaining security.
Mr Ivanko said Unmik would keep a small legal office to "assist Kosovo in dealing with non-recognising countries" such as Serbia.
Unmik will be ready to accompany Kosovo officials to international meetings where some participants refuse to recognise Kosovo, he said.
Unmik will help Eulex in logistics and transport. "Unmik might have some civil affairs office in the Serb-controlled part, but no police," he told the BBC.
Eulex police will go on patrol with Kosovo police, "to act as monitors, mentors and advisers," Ms Limdal of Eulex said.
"It's clear Kosovo institutions will be in the driving seat," she said, but explained that Eulex would take over Unmik investigations into organised crime and alleged war crimes.
Eulex will also work hard on curbing people trafficking and helping to track down some 2,000 people still missing from the 1999 conflict, she said.