If you know your boss is 100% wrong about something, how would you handle this?

If you have garnered proof that your boss is100% wrong the best thing to do is to gather your proof formulate options and then arrange a private meeting with your superior.
Face it, no one wants to hear that they are wrong and going into a meeting without any proof or any suggestions for improvement is setting oneself up for failure.
You should never tell anyone that they are wrong.
Present the facts and let the facts speak for themselves.
Show your superior what you have discovered and how it can hurt the company.
Then show ways that can turn the course and why you think it will work and how it can benefit your superior.
Just leave it as your suggestions.
If your facts are clear and tacit the information will speak for itself.
If your superior is one that believes in growth he or she will consider the information given and later laud you for the catch.
In addition he or she has ideas wherein he or she can implement to turn the tide.
This still empowers your leader and puts you in the light of a problem solver who cares about the company.
Approach is everything.

i ussually called for a private meeting to make a point about the subject mistakes and to sit down with him and talk how to handle the situation...so that it can be avoided to next level....

"Disciplining yourself to do what you know is right and important, although difficult, is the highroad to pride, self-esteem, and personal satisfaction" Margaret Thatcher It depends on the relationship between you and your manager& his or her why of managing staff: Authoritative Boss &the Narcissistic Boss you cannot tell them they are wrong (it’s their way or the high way) The Pace-Setting Boss & the Everyman Boss you can let them know

to define / discuss logic for right decision

since knowledge is built upon facts then the best way in my opinion to face him with facts, and to avoid any kind of embarrassment to him the act will be within closed meeting between me and him .

I believe that a Boss and Subordinate are part of1 team.
Even if the Boss is wrong, it is the duty of the Subordinate to inform the Boss so that external teams do not identify the mistake and hold the same against your team.
I agree with Federico - best way to deal with this is in person without any external presence so as to avoid any discomfort and embarrassment.

say your opinion in professional manner but never challenge them.
its not healthy to be on the other side of your manager, remember always that this is your team and he is part of your team regardless.
we live in world that one need alliance, you either keep peoples to your side or the other side.
I prefer to keep peoples to my side, I still say what I think politely.

1st of all i have to be ensure from my opinion200%2nd i have to tell him cur fully that i think only think that i have another solution and i need you to ensure from my opinion.
then you will tell him the correct and you will win his acceptance.

sometimes we know that something is wrong.
the first step is to form this knowledge in a way which is clear for others.
by means we have to provide compelling proofs to be able to convince others rationally.
and the boss is boss.
i say my idea as a new proposition which has better results and less potential risks rather than challenging about his/her idea.

افوض امري الى الله و اقول "حسبي الله ونعم الوكيل"

Depends first what's your position (unfortunately) and it also depends what's your boss "made of".
If your position is "low", whatever you say and do won't be considered (it's very rare and it's very unfortunate).
I think there are a lot of factors and cannot be generalized.
But i'll make it simple in logical cases.
If the boss is100% wrong and I know it, I'll just fix a meeting with him after work preferably, or have a coffee, and come up with a nice file for him to read, with all details and arguments that prove him wrong.
If with all this he still sticks on his stance, I won't add more.
I would've done my job, but I'll make sure to protect myself by going to higher managers and let them know what happened.
Only then I'll back off.
At the end, time will tell.
He's going to get laid down and I'll get laid up!