I have tried searching the Gnome.org web site to find an answer. I have tried a web search based on Gnome 3 requirements. I am surprised at the number of times that this or a similar question has been asked on the forums of various Linux distributions and the same answer is always given. It is a quote from this page:

Gnome Myths

It is our primary focus to build a modern operating environment, platform, and user experience. It doesn't make sense to target the hardware of the past. GNOME Shell uses relatively primitive 3D capabilities that have been available from essentially all computing devices made in the last four or five years. This includes most desktop and laptop computers, mobile devices, phones, tablets, and netbooks. Where there are exceptions, largely, there are bugs we can and should fix.

So, the official plan is that people can still use the GNOME 2 shell with GNOME 3 applications and libraries, if necessary, but this is a transitional state, and to get the GNOME 3 experience, your computer needs hardware acceleration.

So, the answer from the official source is "you need a computer built in the last 4 to 5 years and to get the Gnome 3 experience you need hardware acceleration." And that means a video card capable of doing accelerated graphics.

Personally, I would not call that quote a proper specification of hardware requirements for running Gnome 3. But there you are.

In your case if you do not intend to fit a video adapter then you will not be running gnome shell or any other user interface. And that is fine. So, the "hardware acceleration" requirement is not appropriate.

My machine is four years old and it has a slightly lower specification than your machine and it is running Gnome 3 with Ubuntu Unity both in 11.10 and 12.04 development branch in a more than adequate way.

But for completeness of answer I repeat my comment that the video card is the key component for running the User Interface on a Gnome 3 desktop environment be it Unity, or Gnome 3 shell. And a video card with anything less 512GB memory will give a disappointing performance unless used in fall-back mode.

Update:

I have found the release notes for 11.10 Oneiric Ocelot which is the first Ubuntu to be built on the Gnome 3 desktop environment.

11.10 system requirements

The minimum memory for the desktop is 384MB. But this comment might be more applicable for you:

The minimum memory requirement for Ubuntu Server 11.10 is 128 MB of memory.

Again for completeness I add this link:

Unity Graphics requirements

If the video card is capable of doing this it is capable of running Unity and I would guess that it is also capable of running Gnome 3 shell:

OpenGL Version required by Unity

Unity requires OpenGL 1.4 of higher. Because of the decoupling between Opengl versions and the first implementation of new extensions by hardware vendors, just having OpenGL 1.4 is not enough. The system must also have support for a number of OpenGL extensions.

OpenGL Extensions required

Framebuffer Object

Rectangle Textures

Non power of 2 textures

Vertex programs

Fragment programs

buffer objects

GLSL shader support

(Optional) Other requirements

Minimum 128MB of video

memory Minimum texture width/height 2048

Enough > registers for vertex and fragment shaders programs

Regards.

In addition to Wilf (I do not have 50 reputation points yet to comment :s), the official docs that back this claim are found here: http://www.ubuntu.com/internet-of-things (scroll to "Widest range of supported devices")