The aye-aye is a kind of lemur, and looks a bit like a cross between a mouse and a bat.
There are as few as 1,000 aye-ayes left in the world.
At one point they were thought to be extinct due to hunting in Madagascar, an island off the southeast coast of Africa.
Lynsey Bugg, from the Zoo said:  "Their mother was very secretive and we didn't realise she'd had twins straight away."
"It's a world first, so it wasn't what we were expecting but it's amazing, really amazing.
"We wanted to monitor them and make sure they were putting on weight before we announced their arrival."
Aye-ayes have been born in captivity in Britain before.
A girl called Kintana was born in 2005, and a boy called Raz was born in 2008.
But this is the first time that twins have been born.
Lynsey said that the twins' mother Sabrina was doing a "great job with her young".
"The twins are physically demanding and it is testament to Sabrina that she and the twins appear to be doing really well," she said.
"Whilst we are keeping a close eye on the group, we try to adopt a hands-off approach and encourage a family dynamic close to one they would experience in the wild."

Selling animals under 8 weeks old has also been stopped under the new rule
Katy Tang, one of the politicians who helped to bring in the ban, said that she did it because she doesn't think some pet stores spend enough time thinking about animals' health or well-being.
She says sometimes puppies and kittens are not treated as well as they should be and animals can also be sold when they are very young - before they are ready to leave their parents.
San Francisco isn't the first place in America to bring a rule like this in.
A number of other cities in America - like Phoenix in Arizona and Montgomery County in Maryland - also have laws which mean cats and dogs being sold in pet shops must come from rescue centres.
In the UK, pet shops can't sell cats or dogs that are under 8 weeks old and the pet shops have to be given a license by the authorities in their area.
Twin baby aye-ayes have been born in captivity at Bristol Zoo for the first time.
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San Francisco has brought in a new law saying that pet shops can only sell cats and dogs that come from rescue centres.