Yes, this can be done on a trial account (Which I am on now!) and I have my website hosted with MobileMe.
You can set up your own personal domain, or use a MobileMe domain name (web.me.com/username/site/page). Setting up your personal domain doesn't cost anything, all you need to do is pay for the domain name.
TIP: use
GoDaddy.com to purchase your domain name, and use promo-code GB1 to save money at check-out!
To set up a personal domain with your MobileMe account,
see this support document.
To publish to your MobileMe account open iWeb, create your website and click Publish!
If you don't have iWeb, or are using a PC
see this support document for manually making a website that is hosted on MobileMe