I need to kill a PC

I have a situation. It is a common problem that likely many of you have faced. I have a disruptive PC in my 5th edition gaming group. I have a regular gaming group that meets biweekly. 5 players and me, the DM. The campaign has run for over a year. Over the course of the last 3 months the player in question has decided that his character should turn progressively more evil. This started with little things: torturing information out of a goblin spy, stealing from an orphanage fund the characters had collected. Recently it has gotten worse: he helped an NPC summon a demon so the group could 'fight' it and get a better reward, and started working as an assassin on the side. Finally, this last session, he went over the line. He knew that the mansion the players had been sent to investigate for hauntings was full of innocents. It was supposed to be an investigation and clue-finding puzzle adventure.

He burned the mansion down. Some of the NPCs escaped, mostly thanks to the quick work of the other players, but not all of them. The player was shrewd enough to make sure the fire was set in secret...the other PCs, in character, have no idea this stuff is going on. But out of character, everyone at the table was upset, and it left a bad tasting session.

Note: it is the character that is the problem. The player can be dealt with. I know this person, and he's simply playing a bad character concept to the hilt. Do not tell me how to manage the group. Do not lecture me about how to improve relations between the players. Do not tell me to kick the offending player out of the group. Do not tell me why it is wrong for a DM to kill a PC. The issue is settled. He's going to die.

Problem: if I just fudge dice rolls and target the PC, in addition to being obvious, the other characters will do their duty and heal him, defend him, etc. It would likely lead to a TPK. Nor do I want the other characters to "find out" about his deeds and turn on him. The one time I saw PCs really go after each other, it was the end of several friendships.

So he has to be convinced to kill himself. To take some risk so stunningly stupid that his death is fair, and inevitable. Luckily, the character concept - a bard4/warlock6 greedy for magical knowledge - should make this possible.

Help me kill him. Help him kill himself. This PC must die.