Parents of newborns who share a birthday with the new prince or princess, due later this month, will have to register the birth of their child on The Royal Mint's Facebook page to receive one of 2,015 silver pennies.
The coin will be minted with the one penny's Royal Shield design and the new effigy of the Queen, which was unveiled in March.
It will be presented in a special white box which will be tied with a purple ribbon and which features the image of a silver carousel horse.
A spokeswoman for The Royal Mint said: "It is a fitting gesture to welcome the eagerly anticipated future prince or princess and also in keeping with an age-old tradition to mark a new birth with a gift of silver for good luck."
Thousands of commemorative coins were struck to celebrate the arrival of George and sold out within days.
The Royal Mint produced 10,000 solid silver crown-sized £5 coins, which cost £80 each, and 2,013 22-carat gold sovereigns, which cost £800 each. Both featured the prince's namesake St George.
Babies born on the same day as George were entitled to receive one of 2,013 free lucky silver pennies, and when he turned one the occasion was marked with a commemorative £5 coin.
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