How do I use a custom bit.ly URL in a tweet, rather than t.co?

Question

I've setup bit.ly to use a custom URL. Works perfect if I paste a link into bit.ly, it outputs the custom URL, so I know that piece is setup.

The main question is this: How do I setup the "Tweet" button on my site so that when a user clicks "Tweet" the URL that appears in the tweet is shortened with our custom URL rather than Twitters t.co URL?

Thanks!

MORE INFORMATION:

Just to give a better example: http://bits.blogs.nytimes.com/2009/12/16/the-times-and-bitly-roll-out-nytims-short-links/

Scroll down to the end of the post and click the "Twitter" button, you'll see they are using their http://nyti.ms (http://nyti.ms/NUHtlN) which uses bit.ly...

Answer 1

"All URLs regardless of their length are now automatically wrapped with t.co"

Official Source:

https://dev.twitter.com/discussions/2806

EDIT: Demonstration follows:

Going here, and click the Twitter share button pops up a share box, populated with:

Isaac Upgraded to Category 1 Hurricane http://nyti.ms/RUs5HF

When this is posted to Twitter, it appears that Twitter creates a new wrapped hyperlink, in the form of:

<a href="http://t.co/wToOFYQc">http://nyti.ms/RUs5HF</a>

The Twitter post then has the appearance of using the nyti.ms link, but in actuality, it's just a t.co link.