Brilliant! I was wrong.

One quibble — you can’t actually sample N(0, 1) or any other distribution supported on the whole real line in finite time. But that isn’t a necessary feature of the algorithm — any dense distribution will serve. For example, roll a die and write a minus sign if even. Roll a pair of dice repeatedly. Subtract 2 and write the digit, over and over, until you roll double sizes. Write a decimal point. Do the same thing again, only this time write a 0 when you get double sixes, and you’re done.