The Duchess of Cambridge gave birth to a son on Monday afternoon in London's St Mary's Hospital.
The mint in Llantrisant, Rhondda Cynon Taf, announced earlier this month that children who shared the birthday would be able to receive one of the coins.
It has struck 2,013 of the coins. Parents wishing to claim one need to register on the Mint's Facebook page.
Shane Bissett, director of commemorative coin at the Royal Mint said coins had been used to celebrate special events for over 1,000 years.
"We're honoured to be able to mark this momentous occasion in our own way, which is in keeping with an age-old tradition to mark a new birth with a gift of silver for good luck," he said.
"We send our very best wishes to their Royal Highnesses The Duke and Duchess of Cambridge and extend this to new parents across the country today as they celebrate their own special arrivals."
The coins are worth Â£28 each.
In order to claim, parents need to visit Facebook.com/theroyalmint and register their child's birth within 60 days. They will also need to provide their child's birth certificate.
Babies born on the same day as the new prince are to receive a silver penny made by the Royal Mint.