Does the add-in video card have more than one connector ? Perhaps
two DVI-I connectors on it ?
If so, you can use a DVI-I to VGA passive adapter, and avoid the
headaches.
I'm using one of these right now, to drive a VGA monitor from a dual
DVI video card.
http://images17.newegg.com/is/image/newegg/82-203-020-S01?$S640W$
This is what my video card faceplate looks like. The cross shaped thingy,
with the four dots, is where the VGA analog signals are available. The
passive dongle adapter, picks up RGBHV from those connections, and
puts the VGA on the familiar 15 pin pattern.
http://images10.newegg.com/NeweggImage/productimage/14-130-281-02.jpg
More details on DVI-I are available here.
http://en.wikipedia.org/wiki/Digital_Visual_Interface
*******
Now, to the ugly part. You'd provided no details, as to model number of
the Dell computer, and the make and model number of video card.
On AGP based motherboards, it was popular to disable the built-in video,
as soon as an add-on video card was detected.
On some PCI Express systems, they chose to do the same thing (although
they didn't need to). This disabling is at the BIOS level and may be
totally automated.
Modern PCI Express systems, can accept having both the built-in running
at the same time as the add-on video card. That allows support for
more connectors. With some caveats, as to how Windows may treat multiple
monitors and cards/GPUs.
So, yup, sometimes it works. But probably with a good deal of swearing
along the way

That's why I'd recommend the dongle adapter, if your
add-in video card has multiple connectors available.
There are very few deceptive video card designs, but you should know they
exist. There were a couple, their price point is below $50, they have
both a VGA and a DVI connector. But internally, the video card only
has one interface enabled. That prevents both connectors from being
used at the same time. While this is an obscure failing of a couple
cheap video cards, you should be warned that there are designs that
won't actually support two monitors when you want it. And it means
I have to type up this paragraph as a warning... Most other cards,
like the one pictured here, can drive any two of three connectors.
http://images10.newegg.com/NeweggImage/productimage/14-130-281-02.jpg
*******
If your Dell is a "slim" or "SFF", the video card can look like this.
http://images10.newegg.com/NeweggImage/productimage/14-129-063-01.jpg
And then you're kinda screwed... Those small computers force you
to do it the way you're trying to do it right now. Unless you like
the crappy signal from the round DIN connector.
If you're faced with this situation, you can try looking in the BIOS,
to see if there is a setting for the built-in video (enable/disable),
but I doubt it's there.
********
Now, on to more expensive solutions.
1) Replace the video card, with a multi-head one. They make low profile
multiplexed video cards. They use a "Y" external cable, to make the
two connections needed.
You can tell you're getting the right kind of low profile card, because
it comes with the "Y" cable. The monitor type may be fixed at two DVI-D
or two VGA. Generally, the cable doesn't support any arbitrary kind of
connections. There are limits.
http://images10.newegg.com/NeweggImage/productimage/14-129-106-07.jpg
The card is low profile and has a DMS-59 connector, which has enough
pins to carry the signals for two displays.
http://images10.newegg.com/NeweggImage/productimage/14-129-106-03.jpg
If you look hard enough, you can find "short" low profile cards, to fit
the tiniest of stinking slim/SFF packaging. Expect to pay a lot more
for this solution, than a regular video card.
2) Get a Matrox DualHead2Go or a TripleHead2Go.
http://www.matrox.com/graphics/media/pdf/products/en_gxm_datasheet_intl.pdf
The Matrox product takes a 2560x1024 image from the computer, and
makes two side by side 1280x1024 images from it, driving two monitors.
The "triple" version splits the image three ways. It's expensive and
an active conversion technology. I don't know if it's still for sale.
To the operating system, the operating system thinks it is driving
one monitor, but the image is split between two monitors
electronically.
HTH,
Paul