Can I install Fedora 18 on my computer with a Pentium 4 1.8 GHz Processor, D845GLVA motherboard and 1 GB RAM

Try it and find out.

I've got it on a P4 of some sort with a Gig and a radeon 9600 (??). Works fine last I looked.

Why don't you add 1GB Ram to your computer? And try other distributions like Lubuntu Or puppy Linux?

You should be able to put F18 on a 1.8 GHz P4. Been there, done that, went with the LXDE spin.

This computer has the same speed AMD CPU and the same RAM. When I reviewed F18 I had a lot of problems — horrible new installer, bugs in Gnome fallback, video players unhappy with my hardware — but it installed and ran.

Nevertheless, you'd probably get better performance with something else. Fuduntu was forked from Fedora: for me, it's kept the good things and avoided the bloat and bleeding-edge uncertainty. Mint Mate edition is pretty good, too.

I have Fedora 18 Gnome 3 desktop running on a P3 550 MHz, 784 MB RAM with a nVidia 6200 graphics card.
The MATE DE is also installed and runs when selected from the login manager and believe it or not with
MATE running and having "compiz" and related packages installed the "desktop cube" and "emerald" window
decorator function. No this machine will not set any speed records for sure but is fun to run and test
Linux OS's on. Don't be afraid to try running Fedora and a livecd/usb would be a good way to check out
how it will or will not run.