Legislation to ensure the Duke and Duchess of Cambridge's baby becomes monarch – whether a boy or a girl – is to be introduced as rapidly as possible, Deputy Prime Minister Nick Clegg said.

The 15 nations where the Queen is head of state yesterday gave their final consent for the Government to press ahead with the landmark bill to end the centuries-old discrimination against royal women.

Mr Clegg said: "The Government will soon introduce the Succession to the Crown Bill which will make our old-fashioned rules fit for the 21st century.

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"It will write down in law what we agreed back in 2011 – that if the Duke and Duchess of Cambridge have a baby girl, she can one day be our Queen even if she later has younger brothers."

The development comes as Kate began a second night in hospital where she is receiving treatment for a severe form of morning sickness.

William was yesterday again at the bedside of his wife, who is being treated at a private central London hospital for dehydration.

Kate is "continuing to feel better" and she and the duke are immensely grateful for the good wishes they have received, St James's Palace said.

The duchess is likely to be taking anti-sickness tablets and be on a drip so she can receive fluids intravenously to combat the effects of dehydration caused by a condition known as hyperemesis gravidarum.

Her illness could indicate she is having twins as mothers carrying two babies have a greater chance of developing severe morning sickness.

The world's media are camped outside the private King Edward VII Hospital waiting for any news about the future queen and her unborn child.

St James's Palace also stated that, as her pregnancy was in its very early stages, she was expected to stay in hospital for several days and will require a period of rest thereafter.

Moves towards constitutional change gathered pace in the wake of the duke and duchess's wedding in April 2011 in anticipation they would produce offspring.

In October 2011, David Cameron announced the Commonwealth countries where the Queen is head of state had agreed to give female royals the same rights of succession as their brothers. Since then, the New Zealand Government has been co-ordinating formal consent from each realm.