Fields showing up twice - once where I put them, once where I didn't

I've added a field to the story content type called pullquote (using CCK), and edited the story template to drop the pullquote in in a little box set to float:right, so that it's embedded in the body text the way you see pullquotes used in magazines. Worked OK, with a little bit of experimentation.

However... I then noticed that it was also putting the pullquote in at the very end of the story, just as plain unformatted text. After doing a bit of research, I went into the displayed fields area of the content types admin and changed pullquote from default to hidden -- but that removed it from where it was supposed to be as well!

It seems to be some sort of all-or-nothing situation -- I can either have my pullquote displayed the way I've indicated it should be in my template, but also have it tacked on at the end as well, or I can try to hide it and have neither. This doesn't seem right -- presumably the whole point of using CCK and Contemplate is to be able to have your extra fields display the way you want them to! Am I missing something? How can I have it where I put in the template, and only there?