Here's what I found for a Dell Optiplex GX-620:
Processor: Intel Pentium D 326 / 2.66 GHz 64-Bit
RAM: DDR2 SDRAM - Non-ECC 533MHz (512 - 4Gig)
Drive I/O: SATA-300 (80GB, 7200rpm)
Ethernet: Broadcom BCM5751
Other: ACPI 1.0b , APM (K/B & Mouse are both USB)
Please note, this kind of info is what is needed to give better advice or help troubleshoot. So if it's wrong then please post what you really have.
The quick answer to your question is that you probably can't run a 64-bit virtual OS since you don't appear to have a multi-core processor. You appear to only have a single core Pentium. And even though it's 64-bit (which I tend to doubt), your host OS might only be able use it like a dual-core 32-bit CPU. But it's still just a single core processor and therefore not really able to devote even another "real" core to a guest. VMWare may be able to get around this limitation but I doubt it. And even if it could it's still somewhat pointless. (Read on...)
That said, I will also ASSUME you have enough memory? You can get by with 1GB or even 2GB of system RAM these days. But to really run a 64-bit OS you pretty much have to install a minimum of 4GB - at least for Windows, anyway. Unfortunately, 4GB appears to be all the RAM your "real" machine can even handle. So I hope you're not sharing any of it with an embedded video adapter or something. Then again, maybe there's a BIOS flash for your machine that will allow more RAM - ??? But when you start talking about a virtual environment and the desire to run a 64-bit OS there too, I just have to ask: are you serious?! I'd say don't even think about it unless your real system had at least 8GB. But then again, we don't really even know what you have, do we?
You will also probably need to configure your BIOS so that it passes all the RAM to whatever host is booting. That way your guest OS can use half or a quarter of it without "red-lining." There may also be a BIOS setting with regard to "VM optimization" that you'll probably want to enable. But again, I can't say if that's an option since I don't know what BIOS features your particular model has. Of course, the only way to find out is to look.
Next step would probably be to determine what your host OS can do. You didn't say if your host OS Ubuntu 11.10 was a 64-bit or 32-bit version. If you just downloaded/installed whatever version Canonical had then it's probably just 32-bit. Therefore you're probably out of luck on VMWare being able to run a 64-bit guest OS. Then again, it could be time to switch to Oracle's Virtual Box since I seem to recall VMWare making you pay for 64-bit support. (Maybe I should have started with that.)
All that aside, I am curious why you want to run a 64-bit Windows OS virtually. Like I said, you pretty much can't even use a 64-bit OS to it's true potential unless there is a minimum of 4GB of RAM (and a 64-bit core). To me, it makes no sense. You might also be setting yourself up for even more pain if you're trying to play high end games in a virtual Windows environment too. But if all you're trying to do is run something like Microsoft Office (and can't figure out how to make it work with Wine) then using a 32-bit Windows guest OS is probably all you need anyway - and probably overkill at that!