Is there a device I can buy that will convert the analog signal to digital?
You may find that a USB-to-DVI video adapter costs less than either a new monitor or a full-on VGA-DVI converter.
This works best if you are using the laptop's external display primarily for office work, web browsing, or other non-intensive graphics applications. I would not recommend it for 3-D gaming or for watching movies.
Yes, there seem to be Adapters available.