...For Very Large Values of 1

I was introduced to the following 'proof' years ago. It's become a favorite to share with algebra classes.

a = b
a² = ab
a² - b² = ab - b²
(a + b)(a - b) = b (a - b)
(a + b) = b
(b + b) = b
2b = b
2 = 1

While the result is obviously wrong—one is not equal to two, even for very large values of one—the steps seem logical enough. Justifications for each step are as follows:

1. Define a and b to be equal to each other.
2. Multiply both sides by the same thing (a).
3. Subtract the same thing (b²) from both sides.
4. Factor both sides.
5. Eliminate a common factor (a - b) from both sides.
6. Replace a on the left side with b, since they were defined in step 1 to be equal.
7. Express the sum (b + b) as a product (2b).
8. Eliminate a common variable (b) from both sides.

Looks sound enough, except for that one nagging little detail—two is still not equal to one.

A No-prize to the first person who can tell me specifically where and/or how this 'proof' breaks down.

Edit:
Wendy gets the No-prize:

Removing the common factor in step five, (a-b) is removing the zero...

Specifically, it's removing the zero by division. And dividing by zero (like traveling faster than light or electing third-party candidates) is something you just can't do.