Today, Mozilla has introduced a new version of Firefox that comes with Bing as default and includes the Bing Awesome Bar.
The company says:
Now Firefox users who are Bing enthusiasts can use Firefox with Bing to use the Web the way they want without having to take extra steps to navigate or customize their settings to Bing.
Firefox with Bing offers the latest version of Firefox with Bing set as your home page and the default setting in the search box and AwesomeBar (where you can also type in queries as well as Web addresses). If you already have the latest version of Firefox, then you just need to download the Bing Search for Firefox Add-on to set the same preferences.
The new version, optional to those who want it, is available here: Firefox with Bing