The Dell Latitude D620 is one of a few dell computers in that range (The D820 has the same issue) that have a defective video card module. Basically they used a crappy voltage regulation component in the video card on the units with the Nvidia chipset. The class action lawsuit related to that can be found here: http://en.community.dell.com/dell-blogs/direct2dell/b/direct2dell/archive/2010/09/30/nvidia-gpu-update-nvidia-class-action-lawsuit-and-limited-warranty-enhancement.aspx
Chances are, the video card slagged itself. We had a fleet of 60 D820's at my other job, and we had about 5 or 6 of those go bad towards the end of their lifespans in that same manner. THe other way they sometimes go is just a bad motherboard or power inverter board. It is theoretically possible that the ram chips are loose but still partially plugged in, causing a short.
The good news about the latitude series is that parts are still readily available even for the D620's. The video module is basically grafted onto the motherboard, so you would be looking for a new motherboard either way. $80 for that here (http://www.discountelectronics.com/index.php?l=product_detail&p=4287)
May also want to check here: http://www.parts-people.com/index.php?action=category&start=24&id=140&subid=190
It could also theoretically be the LCD or the LCD backlight, or the backlight inverter, but the problem description with the powering off is more consistant with the motherboard issue.
Make sure you get the same Motherboard you already have, or if you get a different one, ou get the correct matching heat sync. They are different and will not line up otherwise.
You might also look at upgrading. A D620 is 4 generations older than the latest releases now (D620, then D630, then E6400, then E6410, then E6420) You should be able to find at least a D630 somewhere for not much more than the cost of a new motherboard. Newegg has them recertified for $159. http://www.newegg.com/Product/Product.aspx?Item=N82E16834200139
. Finding a 6400 is probably going to be closer to $300-$450, and then up from there. A D630 is about 60% faster than a D620 simply due to the additional cores and processor horsepower. an E6400 is going to be several orders of magnitude faster. The E6410 and E6420 are actually pretty similar in performance, but either one will be very expensive. a D630 gets you into "will run windows 7 without exploding" territory.
I was using an E6500 for my work laptop up until 2 weeks ago, they are nice machine. (the E6500 is basiclaly a slightly bigger e6400).