I got the usual story: $17/record and you have to be related to the person. Since this is a birth record from over a hundred years ago, this it-isn't-public thing sort of makes me a little nuts, but I'm polite when I'm trying to get something (generally), and got the Coolest Response Ever.
Me: "So I can't get it, even if it's over a hundred years old?"
Her: "But you are descended."
Me: "She's my great aunt."
Her: "That's fine. You have to show the relationship but you can go on ancestry.com and just print it out."
Lesson learned: don't take these things too literally, and life is easier all around.
And then it got better. She was quite happy to check a particular date/name combination on the phone and confirm it wasn't there. So I don't even need to spend the $17 and I got an immediate answer.
Still leaves me with the puzzle (possibly unsolvable) of _which_ county in Colorado, but this is a relatively low-priority question anyway.