It is understood that Kate's pregnancy has not passed the 12-week stage - a similar position to when she was struck by the illness when in the early stages of her first pregnancy.
A Kensington Palace spokesman said the Duchess's attendance at future events would be decided on a 'case by case' basis.
Kate is scheduled to be among the guests at the opening ceremony of the Invictus Games in London on Wednesday evening along with William, Harry and the Prince of Wales and Duchess of Cornwall.
Leader of the SNP Alex Salmond has congratulated the Duke and Duchess of Cambridge after it was announced they are expecting their second child. Mr Salmond used their Scottish titles to send his 'best wishes' to the couple.
Congratulations & best wishes to the Earl & Countess of Strathearn. Wonderful to hear they're expecting their second baby – very happy news!
Labour leader Ed Miliband says it is "fantastic to hear that Prince George will soon be a big brother" following news the Duke and Duchess of Cambridge are expecting their second child.
Fantastic to hear that Prince George will soon be a big brother! Congratulations to the Duke and Duchess of Cambridge on their happy news.
The new royal baby will be born fourth in line to the throne. As a sibling to Prince George, the Duke and Duchess of Cambridge's second child will not be expected to be crowned sovereign.
But second-born royal children - often dubbed the 'spare to heir' - have on occasion ended up as monarch.
The country's last king, George VI, was not meant to accede to the throne and only did so when his older brother Edward VIII abdicated over his love for American divorcee Wallis Simpson in 1936.
Kate's first pregnancy was revealed when she was just a few weeks pregnant after she was admitted to hospital suffering from severe morning sickness in December 2012.
Hyperemesis gravidarum, which affects 3.5 per 1,000 pregnancies, causes severe vomiting and can lead to dehydration, weight loss and a build-up of toxins in the blood or urine, called ketosis.
Prince George is likely to be joined by a little brother, according to the latest odds from bookies.
Ladbrokes make it 4/5 that the Duchess of Cambridge gives birth to another boy in early 2015.
James is the early favourite to be the name of choice at 6/1, with Arthur and Victoria following just behind at 8/1.
Charles, Thomas and Alexandra are also high up in the odds.
Early predictions on Royal births are not always accurate - the odds on George were 10/1 prior to the future King's birth last year.
Kate was due to join William in formally opening Oxford University's £21 million centre dedicated to the study of China. He will still attend, as planned, Kensington Palace said.
The announcement that the couple are expecting their second child comes two months after Prince George turned one. The third-in-line to the throne was born on July 22 last year to a worldwide fanfare.
Kate suffered with hyperemesis gravidarum with her last pregnancy and is suffering from the very acute morning sickness again, meaning she may need extra hydration, medication and nutrients.
The couple's second child will become fourth in line to the throne, shifting Prince Harry further down the line of succession.
The Duchess of Cambridge's announcement will throw into doubt whether she will be able to attend her first solo overseas tour, set to take place in Malta later this month.
Kate and William's second child will join second-born Royal children as being known as the "spare to heir".Read the full story ›
The Duchess of Cambridge is suffering from severe morning sickness, the palace has said, after announcing her second pregnancy. The Duchess was admitted to hospital during her first pregnancy when she was carrying Prince George
The Queen is said to be 'delighted' by the news that Prince William and the Duchess of Cambridge are expecting their second child.
The Duchess of Cambridge will not be attending a planned engagement with the Duke of Cambridge in Oxford today as she is suffering from acute morning sickness. Clarence House announced news that the couple were expecting their second child this morning.