In one sense of the word it is impossible. Probably not to prove it in a technical, evidential sense. But we must realize that if JC himself stood before us and said Hussein was ineligible, 40% of the people would not believe it. And a number of others would not care anyway.
So, what it comes down to is this. What will we do if it is actually proven he is not legally President and our governmental legal system refuses to depose him?
This statement is a catch 22. It can not be legally proven without the courts.
Then it could turn into much like the impeachment of Clinton where he did brake the law but it wasn't a serious enough crime to remove him from office. I don't know how they could twist it to make someone that can not even qualify to be president to some how qualify.
Who knows maybe that decree the congress set a couple days ago officially makes him a 'citizen' but not a natural born citizen and in heir collective minds this is not strong enough evidence to remove him from office.
Personally, I believe the last time for a procedural intervention was after the elector college and before being sworn in. Some might see the house having jurisdiction over the president without impeachment but there is a difference between a president elect and a president. Then impeachment becomes a political question and right now that would not be in the cards. Could you imagine the race relations if we impeached a black president? The first black president?
This birth issue, real of imagined, will wax and wane in proportion to Obama's poll numbers. The more he goes down, the more this adds fuel to the fire and the birther's look smart. The more his polls go up the more the birther's look like kooks. This issue can help take away his political capital and it would add fuel to the fire over another issue, but in an of itself, it will not remove him from office.