Yooranah was born at Edinburgh Zoo in May 2013 but keepers waited until he had fully emerged from mother Alinga's pouch before they started to handle him.
He was only the size of a jelly bean, hairless and blind when born but now weighs 1lb 6oz (632g) and has excellent vision.
Keepers have started a routine of regular weigh-ins, during which Yooranah is placed on a plush toy koala from the zoo's gift shop, to give him something to cling on to for stability as well as comfort.
As he is still so small, the keepers use a digital scale similar to those used to weigh baking ingredients.
He will move on to the adult's scale when he is much larger.
Donald Gow, senior keeper for koalas, primates and hoofstock at Edinburgh Zoo, said: "It's very exciting to watch Yooranah as he grows.
"This is the first time that a koala joey has been born in the UK so it is amazing to see him develop from being pink and hairless into a fully-formed, if tiny, koala.
"At the moment he seems to be all ears! He is also starting to develop his own personality - he can be quite feisty like his mother Alinga and father Goonaroo."
Yooranah is an indigenous Australian word meaning "loving".