How do I keep Firefox from prompting for username/password with HTTP Basic Auth with JQuery AJAX?

I'm writing some browser side dynamic functionality and using HTTP Basic Auth to protect some resources. The user experience is very important and is highly customized.

Here's a simple test JQuery method that eventually will test if a user has supplied the right credentials in a form:

$(document).ready(function() {
    $("#submit").click(function() {
    var token = Base64.encode($('#username').val() + ':' + $('#password').val());        
    $.ajax({
      url: '/private',
      method: 'GET',
      async: false,
      beforeSend: function(req) {
        req.setRequestHeader('Authorization', 'test:password');
      },
      error: function(request, textStatus, error) {
        if (request.status == 401) {
          alert('401');
        }
      }
    });
    return false;
  });
});

If they are not allowed to access /private, at the moment they should see just the alert box. However, on Firefox, a browser-provided login form pops up (to retry with new credentials). Safari does not do this.

We want to completely control the experience with custom forms, fades, transitions, etc. How can I keep Firefox's default box from being shown? (If this will be an issue when we test for IE, I'd love to hear solutions there, too.)