Golfers Laura Davies, Renee Powell, Belle Robertson MBE, Lally Segard, Annika Sorenstam and Louise Suggs also accepted honorary member invitations.
George Macgregor OBE, captain of the club said it was a "historic" day.
In September 2014, the Royal and Ancient voted in favour of allowing women members for the first time in its 260-year history.
More than three quarters of the club's 2,400 members voted with an 85% majority.
"It is an honour and a privilege for the Royal and Ancient Golf Club to welcome these remarkable women as honorary members," said Macgregor.
"The Princess Royal enjoys a strong bond with Scotland and has shown great energy and commitment to developing sport through her work in the Olympic movement.
"They are extremely worthy additions to our roll of honorary members and will become ambassadors for the club as they have been for the sport of golf throughout their careers."
Analysis - BBC golf correspondent Iain Carter:
The Royal and Ancient Golf Club have identified many of the great and the good of women's golf with this list of new honorary members. By selecting the likes of Dame Laura Davies and Annika Sorenstam, it is clear they want to showcase the fact that it is now a mixed club.  This is vital for its role as a governing body, although it does remain one of the most exclusive of golfing establishments.
The Royal and Ancient has been the guardian of the rules of the game since 1754, although in 2004 it devolved responsibility for the administration of the game and the Open to the newly-formed R&A.
Royal and Ancient's members play on the St Andrews links course in Scotland regarded as the "home of golf".
Before the vote to allow female members, women could play on the course, on Scotland's east coast, but they were not allowed in the clubhouse.
They also had no significant part in the sport's rulemaking arm of the R&A.
A statement from the R&A added: "In addition to the honorary members, a number of women have been admitted as members of the club with more set to follow in the coming months."
The Princess Royal has become one of the first women to join the Royal and Ancient Golf Club of St Andrews.