The favicon for Joomla 1.5 templates is kept in the templates/template_name folder. The file is called favicon.ico and you can easily change your favicon by replacing this file with your new favicon file.
You can create a favicon manually on your computer or you can use an online favicon generator. Here are several such generators:
Once you create the favicon, you should upload it to templates/template_name and rename the file to favicon.ico, replacing the old one (it is recommended to keep a copy of the old favicon, just in case you wish to restore it at a later moment).
Clear up your browser's cache and your Joomla's cache (if it is enabled) and reload your website. The new favicon will appear in your browser's address bar 
If you have problems changing your Joomla favicon yourself, ask your host for assistance. SiteGround provides the best Joomla hosting and such requests are handled with ease.
Joomla is a powerful software that is easy to work with once you have the right host to support you. Before you lose any more time with the wrong host, check out the SiteGround Joomla hosting offer.
