Unless you have changed the default path your favicon should be uploaded into your template's images folder as per line 9 of rt_head_includes.php.
As Yves says, it is sometimes very difficult to get IE to pick the new favicon up so your best bet is to try in FF first to make sure it's being picked up at all. If it is then what I generally do is delete any existing favicons from the images folder, delete any bookmarks I have to the site, empty my cache and then re-bookmark. Even then it can take a while before IE recognises the new favicon. As long as it displays in FF I'm happy that visitors will see it even if my IE won't.
I guess it just takes actually closing the browser as I have cleared the cache 5 times and it still doesn't come up. But when I use Safari, it comes right up. BTW, to clear the cache in FF on a mac simply press F5. Or you can select Tools>Clear Private Data then simply select cache's check box and press "Clear Private Data Now".
I downloaded Phoca Favicon (which is a really quick and easy component).
Clicked Create Favicon
Choose Template (which is rt_chromatophore for Joomla 1.5)
I choose the image I uploaded
And Clicked Create and that's it.
So, I went to preview my site
http://www.TheLandman.net
using the Firefox browser. And it still had the Rocket Theme favicon. So, I refreshed the page. It auto-loaded the Rocket Theme Favicon and show My Favicon for a split second, then ended up having the Rocket Theme Favicon. I refreshed a couple time, plus used different computer and still the the same result. What would be the best way to fix the problem?
in your site thelandman.net
i viewed the source and found: <link href="/templates/rt_chromatophore_j15/favicon.ico" rel="shortcut icon" type="image/x-icon" />
<script type="text/javascript" src="/media/system/js/mootools.js"></script>
<script type="text/javascript" src="/media/system/js/caption.js"></script> <link rel="shortcut icon" href="/templates/rt_chromatophore_j15/images/favicon.ico" />
remove one of the line in that pointing to the favicon