The “classic” way to do this is to use a background image on a div around the text field and not setting the border / outline of the text area to none, for example:
<div style="background-image: url(image/with/rounded/corners.png);">
<textarea style="border: none; outline: none;">
</textarea>
</div>
CSS3 supports rounded corners, but this is poorly supported, see for example http://www.css3.info/preview/rounded-border/
EDIT: The answer was correct at the time, but border-radiusin IE is only supported from version 9.
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