Beta(0,1) is an improper prior. Do you mean Beta(1,1), the uniform prior?
In that case, it's a silly prior. You can't be certain that it's exactly that probability.
If that's what you're using, you'd have a 50% chance of getting heads if you were wrong, and a 0.0001% chance if you were right, so:
P(pi = 0.000001) = 0.99
P(H) = P(H | pi = 0.000001)P(pi = 0.000001) + P(H | pi != 0.000001)P(pi != 0.000001)
= 0.0000010.99 + 0.50.01
= 0.00500099
~= 0.005
P(pi = 0.000001 | H) = P(H | pi = 0.000001)*P(pi = 0.000001)/P(H)
= 0.000001*0.99/0.005
= 0.000198
So there's a 0.02% chance that you were right about there being a 99.9999% chance of heads. Essentially, you can ignore that. You now have a 66.7% chance that the coin will land on heads, with much the same distribution as if you started with a uniform prior.
If you meant Beta(1,2) the answer is similar.
Edit: How do you make an asterisk show up, instead of italicizing?
Use a backslash to escape it; for example: * (which is \ followed by the *)
This isn't intended as a full discussion, I'm just a little fuzzy on how a Bayesian update or any other kind of probability update would work in this situation.
You have a coin with a 99.9999% chance of coming up tails, and a 100% chance of coming up either tails or heads.
You've deduced these odds by studying the weight of the coin. You are 99% confident of your results. You have not yet flipped it.
You have no other information before flipping the coin.
You flip the coin once. It comes up heads.
How would you update your probability estimates?
(this isn't a homework assignment; rather I was discussing with someone how strong the anthropic principle is. Unfortunately my mathematic abilities can't quite comprehend how to assemble this into any form I can work with.)