It is possible.
http://www.ascap.com/weblicense/
The minimum fee is $288.00 per year. They will also want proof of your financials, revenue statements, and so on. Because they usually get a percentage of total revenue.
I would advise against pursuing this, though.
Find some original music that you can license, library music, a local band that matches your "sound" and thinks it would be cool to promote themselves that way.
If you try to do this (license a popular song), approval will be required not just of the publisher (ASCAP/BMI/SESAC), but also of the writers and possibly the performers.
There are several types of licenses depending on what you envision using the song for.
The one I linked is for experimental webcasts, and limits you to using :60 seconds or less in one intro or movie scenario.
Major pain in the ass, IMHO.
If you go forward, please post and let us know your experiences.
Also, as way of warning, if you DO this, they WILL check your site.
And if you have any other unlicensed material, even by mistake, they can and will take legal action. If you have a movie scene that has a radio in the background, and they hear a song over that radio, they will come looking for more money.