The priest who was shot in France last week had been having an affair with the prime suspect's wife, according to prosecutors.
He was left seriously injured and was not able to speak to investigators until he emerged from a coma on Wednesday.
It was initially feared the attack could be terror-related as it happened a few days after three people were stabbed to death at a church in Nice.
However investigators have now ruled out that motive after arresting a 40-year-old man on suspicion of attempted murder.
"He turns out to be the husband of a woman who was having an affair with the victim," said the public prosecutor Nicolas Jacquet in a statement.
The suspected gunman made a "full confession" after being detained on Friday, Mr Jacquet added.
He is a Georgian father-of-two who lived near the church, according to Le Parisien newspaper. His wife, said to be a 35-year-old Russian, was also arrested.
The prosecutor said the investigation "will continue to establish the motive and the exact course of the facts".
Mr Kakavelakis had served as a priest in the church in the 7th arrondissement of Lyon for 10 years but submitted his resignation last month.
Register for free to continue reading
Registration is a free and easy way to support our truly independent journalism
By registering, you will also enjoy limited access to Premium articles, exclusive newsletters, commenting, and virtual events with our leading journalists
Already have an account? sign in
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies