Hmm OK if the pc is from 2003 or so, you probably have a VERY early version of pci-e slots - probably 1.0a. From reading about the pci-e versions and the HD 6450 specs
http://www.amd.com/us/products/desktop/graphics/amd-radeon-hd-6000/hd-6450/pages/amd-radeon-hd-6450-overview.aspx#3, a few things
HD 6450 requires:
Minimum 1GB of system memory - presume you have at least this much?
pci-e x16 slot - the mobo slot *is* x16 right? It'd be unusual if it wasn't, but who knows with Dell...
and recommends:
400 Watt or greater power supply recommended - although you had a higher power requirement card work, so your PSU PROBABLY has enough power. But, if it's the one that came with the PC it could be an older ATX level or something so it can't be ruled out completely. If nothing else makes any difference you could TRY it with a higher rated PSU if you have access to one...
Also it is pcie version 2.1. The wiki article on pci-e
http://en.wikipedia.org/wiki/PCI_Express says about pcie version 2.1: "Unfortunately, the increase in power from the slot breaks backward compatibility between PCI Express 2.1 cards and some older motherboards with 1.0/1.0a, but most motherboards with PCI Express 1.1 connectors are provided with a BIOS update by their manufacturers through utilities to support backward compatibility of cards with PCIe 2.1." - so you might want to look for any bios updates available for your motherboard. Now the older the motherboard, the hairier updating the bios was, and having a power glitch or something interrupt the process can make the motherboard unbootable which is a colossal pain in the ass, so I personally almost never update bios except on brand new pc's where there's not much at risk, or unless the bios update looks like it's really really required to fix something. Still, the other card was presumably drawing MORE power thru this same pci-e slot so even this doesn't sound so likely.
One tip I ran across researching the card said "go into the video driver control panel and click "reset to default" - it apparently changes some setting you can't affect through the gui" - I'd probably try that first just 'cause it's the easiest and looks the least risky...
Also you do have the latest Directx installed? If nothing else helps you might try uninstalling and re-installing Directx to make absolutely sure...
And, as usual, it'd be worth running a memory diagnostic overnight just to rule that out - it can be the cause of many a flaky problem. Burn memtest86 to a bootable USB or a CD and let it run overnight...