I read on Wikipedia that Windows 7 on a 64-bit PC needs twice as much RAM as on a 32-bit PC.
I understand why is that: every number stored in memory takes 8 bytes rather than just 4.
That, in simple terms, means that your amount of RAM is reduced to half when you use Windows 7 on a 64-bit computer.

Now, I have a Intel Core 2 Duo Laptop with Windows Vista right now (2 GB of RAM).
My question is:
Since Core 2 is a 64-bit architecture, if I upgrade to Windows 7 will my laptop be working as if it had just 1 GB of RAM?

Or... to say it in other words:
Having a 64-bit PC with Windows 7 do you need twice as much RAM as you need on a 32-bit PC to have the same performance?

If I am right, then I'd say it's a terrible business to have a 64-bit computer and Windows 7 on it (I hope I am mistaken, though).

Follow-up:
After some answers, I'm realizing it's not the same thing to have a 32-bit OS on a 64-bit PC than a 64-bit OS on a 64-bit PC.

Apparently, the problem of Windows 7 requiring twice as much RAM on 64-bit architectures is when you have both the OS and PC supporting 64 bits.
I'd like new answers to address this issue.
Also, is it possible to have more that 4 GB of RAM on a 64-bit PC using a 32-bit version of Windows?