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Political approval from the European Parliament comes after lengthy internal debate on the issue.
The European Commission welcomed the Parliament's support, and said it expected the Council of Ministers to give its support to the move "very soon".
Once the Council of Minsters has signed up to the initiative, the Commission will be able to begin the procedure of choosing a private-sector, non-profit organisation to operate the .eu registry, which will administer the registration of all .eu domain names.
Erkki Liikanen, Commissioner for Enterprise and the Information Society, said that while this new law will provide the basis for the new domain name, "some other measures of public interest still need to be agreed, such as a policy for the extrajudicial settlement of disputes".
Liikanen said, with the support of member states, the .eu domain name "could be available by the end of the year."