A friend of mine had a similar problem with her laptop. It's the power supply on the video card that powers the inverter. The one in her laptop was replaceable, but I'm not sure about your model. Also, they are not inexpensive.
A lot of those older XPS laptops tend to have issues with heating as well.
You do have a few options though. First, external monitor for everything. Not great, but if you have one, you can at least use the laptop. Second, call Dell and get a price on a new video card. The cases are usually pretty trivial to open to replace it, if that laptop does indeed have a replaceable card. You might even try getting in touch with nvidia to see if they can do anything for you. The last options would be looking for something on eBay, Craigslist or local PC shops.
This is a known issue and there was a lawsuit filed against nvidia about two years ago. I'm not sure of the outcome.
From the [http://en.wikipedia.org/wiki/GeForce_8_Series#Problems|Wikipedia article]:
"Some chips of the GeForce 8 series (concretely those from the G84 and G86 series) may suffer from an overheating problem. NVIDIA states this issue should not affect many chips, whereas others assert that all of the chips in these series are potentially affected. NVIDIA CEO Jen-Hsun Huang and CFO Marvin Burkett were involved in a lawsuit filed on September 9, 2008 alleging that their knowledge of the flaw, and their intent to hide it, resulted in NVIDIA losing 31% on the stock markets.
The reason for the high failure rate was because of faulty solder material; when the GPU heated up and then cooled over its use, particularly when used in GPU intensive operations like games, the solder would become soft when hot, and when cool, that solder would crack, causing failure in the 8600M GT, 8600M GS, 8400M GT, and 8400M GS cards. It is widely acknowledged that these cards have a universal design flaw. It has also been speculated that mobile cards prior to the GeForce 8M series like the GeForce Go 7600 and earlier also have this design flaw, although it has not been proven."
I wish you luck!