The Tanaiste has weighed in on speculation over potential names for the new royal baby, saying they should call him Eamon.

There is hope the British public could get a glimpse of the newest royal later. William and Kate have spent the night in hospital with their new baby son whose third in line to the throne.

He was born just after 4pm yesterday and weighed in at 8 pounds 6 ounces. The Prince of Wales says he is "enormously proud", while the US President Barack Obama has wished the royal couple "happiness and blessings parenthood brings".

US talk show host Ellen DeGeneres, recently informed by the New England Genealogical Society that she is a distant relative of the future queen, tweeted her congratulations on Twitter.

"It's a boy! So happy for my cousin Kate and the future King of England," she wrote.

The Canadian Prime Minister Stephen Harper said the arrival of a future sovereign of Canada is a "highly anticipated moment for Canadians given the special and warm relationship that we share with our Royal Family".

Australian Prime Minister Kevin Rudd said "I think all Australians at the bottom of their hearts wish the Royal bub all the best, and certainly wish the new parents all the best as well. When a new bub comes into the world, any old day, any part of the world, it is frankly a time for rejoicing."

He also announced the country's gift to the baby prince - a research project examining the Australian marsupial the bilby.

Meanwhile the arrival of the royal baby will be marked in tonight's episode of 'EastEnders'. Show bosses ordered some last-minute filming to reflect the birth of the future king yesterday afternoon.

A scene showing Dot Cotton and Abi Branning discussing the happy news was filmed last night.

There is no name yet - but the Tanaiste Eamon Gilmore has some thoughts.