The Duke of Edinburgh is receiving treatment for an infection.
Buckingham Palace announced in January the 94-year-old head of state and the Duke of Edinburgh, 99, had been inoculated.
The Duke of Edinburgh is expected to stay in hospital for several days.
The baby boy, born on February 9, is 12th in line to the throne.
The Duke of Edinburgh was admitted to the King Edward VII Hospital in London.
More than a million people have now received their first jab to protect them against the virus.
A small number of guests were also there, Buckingham Palace has confirmed.
The 100-year-old veteran raised more than £32m for health services charities by walking laps of his garden in lockdown.
As countries around the world go into lockdown amid the Covid-19 pandemic, you can visit or enjoy tourist sites and places of interest from the safety of your home. To help cope with your self-isolation, famous places including Buckingham Palace, The Houses of Parliament and Disneyland are offering virtual tours, while the National Theatre in London is streaming plays every Thursday for free.