The pregnancy had already been announced but the due date had not been released.
In a statement, the palace said the duchess continued to suffer from extreme morning sickness but her condition was “steadily improving”.
The duchess will make her first public appearance on Tuesday since her pregnancy was announced in September.
She will join Prince William to officially welcome Singapore’s President Tony Tan as he begins a four-day visit to the UK.
The duke and duchess’s second baby will become fourth in line to the throne, behind older brother Prince George, who was born in July last year.
For the second time, the duke and duchess have been forced to announce a pregnancy before the duchess passed the significant 12-week milestone.
The duchess is again suffering from acute morning sickness – called hyperemesis gravidarum, a condition that may require supplementary hydration, medication and nutrients.
It 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.
BBC royal correspondent Peter Hunt said there had been concerns that the duchess’s symptoms would persist, and there would be “relief” that her condition was improving.
With the new baby behind Prince George in line to the throne, our correspondent added that Prince William might have “a hope – it could be a deluded one – that there won’t be same same intense focus” from the media and public.