What is the correct way of saying that the song grew louder gradually?
His song rose in a crescendo
His song rose to a crescendo.
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Because a song can rise both in pitch and in volume, for clarity you might consider simply using crescendo as a verb:
His song crescendoed.
As for the two options you presented, I would choose "in a crescendo" because the crescendo represents the change in volume, not just the final level. As @mikeagg pointed out, the song may have risen to a climax.
To a musician, crescendo is an instruction to get louder, and a crescendo is the action of getting louder. In this context, rose to a crescendo is nonsense.
However, like many technical terms, crescendo has been adopted into common language, with a different meaning; in this case, something like climax; so rose to a crescendo is a common expression.
Rose in a crescendo is a possible phrase, preserving the musical meaning, but it is not very likely, because the musical crescendo already conveys the sense of increase. I don't think it would be said with the general meaning of crescendo.