I have a computer with an integrated gpu on the motherboard. Because I wanted to use two monitors and my motherboard only has one VGA port, I bought an ATI radeon HD 6450. Both my monitors have a VGA connector, so I also bought an adapter (VGA to DVI). The VGA port on the card works fine, but I can't get the DVI port to work. Does anybody know how I can get this to work?
Hm. From my understanding, you are attempting to connect two monitors to your new discrete card and then use that to manage displays.
If your BIOS outputs to both monitors (which it should, since that seems to be the default with more modern boards), then that confirms that everything is working. In that case, I cannot offer much advice beyond cleaning out the connectors between the monitor, adapter, and video card itself. As well, ensure that Windows is recognizing the monitors in the first place. Either check the AMD Catalyst Control Centre, Windows Screen Resolution configuration, or even check Device Manager to see if they show up. I'm not sure if they show up in device manager, but I do know that Piriform's Speccy does show connected monitors, so I imagine Windows device manager does as well.
However, if your second monitor is completely dead, I would look into checking for hardware failure on the monitor, the video card's DVI port, the adapter, and any cables in between. I personally would replace the adapter, since they seem to have the highest failure rate in my set-ups. As mentioned above, I would still definitely try to clean out the connectors to make sure there isn't a piece of dust in between. But first, you should try replacing the first monitor with the second ("dead") one to see if it even works at all.
While I hold a little doubt the hardware is at fault here, it is entirely possible.. unless you mentioned that "the monitor outputs" somewhere and I missed it.
Try to obtain latest Catalyst drivers and try to detect second monitor(enable) in Display options (AMD/ATI) Display Manager, and also you need check is there any conflicts in BIOS with integrated graphics like Josh already proposed. Also in every manual that comes with Graphic cards is explained a way to connect cabling to multiple displays. Maybe Yours problem is in DVI/VGA adaptor and/or cabling.