To be honest, I can understand why so many would prefer him to be executed. Emotionally, I would like to see him in the gallows for what he has done. But that wouldn't do more than sate our thirst for some form of revenge and to prove his point. He wants to be executed, because it's an integral part of his christian martyr fantasy/delusion. Furthermore, there are people out there who idolize him already, and killing him will only make it simpler for them. Dead heroes can't disappoint, after all.
Really, the best solution, in my opinon, would be to simply toss him into the loony bin, where he belongs. He's not mentally fit, and not only is it his human right to recieve mental care, it's also the only way to both ensure his safety and the safety of society from him.
Some may be shouting: "But that's not a punishment! He needs to suffer, and putting him in an asylum is like putting him in a smart hotel! Why is he being rewarded for his crime?!"
Well, throwing him into an institution -would- be the more severe of punishments available for Breivik. It's the last thing he'd want. It both devalues his cause as well as his bloated self-image. He wants to be killed, both to prove his point about the evil, vile multiculti-dictatorship and to get his role as a martyr confirmed. If he's being put to death, then he and his cause are taken seriously, as opposed to a violent delusion in a mentally ill person's head.
It'd be the ultimate show of disrespect and devaluation to him and his repulsive agenda to treat him and it as a mental case to be treated.
Furthermore, a mental institution isn't some sort of luxurious resort, and a person scentenced to psychiatric care doesn't "get away scot-free". Dangerous mental patients are still kept locked-up and often sedated and on powerful drugs. Also, Breivik will always run the risk of being killed by another patient, or some other assassin. He'll never be safe from someone's need for revenge as long as he live. Even assuming he is ever let free, he's still going to be a target for the rest of his life.
Trust me, a prison or a cell doesn't have to look like a 18'th century dungeon with barren rock walls and moldy bread to be a suitable punishment. Considering everything Breivik have is his twisted fantasy ideology, seeing it for what it is, delusions and hatred, is already a punishment in itself.
Breivik is a sad, sad mental case, and he failed miserably at what he set out to do. He belongs in an institution, reguardless of if he can be cured or not. What truly matters isn't making sure he's miserable enough to sate our natural need for retribution, but to heal the damage he's caused, ensure it will never happen again, and to learn and grow stronger from the experience.
As for silencing him... That would also only prove his point, and it would make clear that we fear what he has to say. And that would give his words much more value than they deserve.