So sit him down and talk to him. Get him to come back to you in a given period (say a week) with his further thoughts, at which time you discuss these and ask him to make a choice.
If he says that he remains convinced he wants to leave, ask him when he’s thinking of, as you now need to start looking. Tell him that as he’s made up his mind, you’re taking this as notification of his intention to leave, which you’d appreciate him following up with a letter. If he’s staying, brilliant. Either way, get him to choose a position.
It sounds very brutal, but I had an employee some years back who kept doing this. He was an essential member of staff, who constantly seemed to dither, and it was unsettling the team. So I forced him to choose….he left, but amicably and at a time to suit us both.