What is the Korean (or Asian) fruit that looks/tastes like melon but is small?

[juliezuck]

A Korean boy in the school where I work (in the US) had small pale canteloupe-colored fruit pieces for lunch. It had been peeled. It was about the size of a small apricot, but round rather than oblong, and firm in texture. He was nice enough to give me a piece. It was quite delicious -- sweet but not cloying. It tasted a bit like melon, a bit like apricot and a bit unlike anything else. He didn't know the name in English (it may not have one) and I don't remember the name in Korean. Any thoughts? It might be some kind of ume plum, but it wasn't tart at all.

[Cookie]

When we lived in southern California we had several plumcot trees among our apricot and plum trees. Some of the plumcot varieties had a canteloupe color, were round, and tasted very much like the fruit you described.

There are also pluots (which are closer to plums) and apriums. But plumcots are more like apricots.

But, of course, the fruit you tasted could be something completely different.