Judging by [this question on rhyming in classical poetry](https://latin.stackexchange.com/q/1036/79), it seems that rhyme did not make its way to Latin poetry in the classical era.
As far as I know, even hymns intended for singing (like *Carmen saeculare*) were not rhymed.
However, rhyming is very common in various songs (and perhaps poetry as well?) in ecclesiastic Latin.
When did rhyme enter Latin poetry or song?
I assume it was gradual, but I don't even have a big picture or an idea of any major milestones in this respect.
For example, what was the first clearly rhymed piece of Latin poetry?

(I'm not sure if this turned out as clear as I hoped. Please ask if this feels too broad or needs clarification.)