A statement said that Pope Francis had agreed with the decision but did not say whether it was the pope's idea that O'Brien should leave for what the Vatican said would be "several months for the purpose of spiritual renewal, prayer and penance".
The statement did not say where the Cardinal was going.
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Outbreaks of rain and strengthening winds will affect southernmost parts of England and Wales today.
An earlier debate on the issue was suspended as an 'expression of support' following the Westminster terror attack.
Five people were killed and dozens more were injured after a car and knife attack in the capital on Wednesday afternoon.