PCIx is backwardly compatable with PCI so, technically, you should be able to plug that PCIx card into a PCI slot and it should work.
PCIe is not compatible at all with PCIx and I doubt you will find an adapter as PCIe is based on serial connections and PCIx uses high speed parallel signals.
If you want to buy a PC with PCIx slots you are looking at server motherboards and their usual need for special memory sticks.
Apart from there were two standard of PCI - either 3.3v or 5v... Most desktop systems have 3.3v PCI slots - but many PCI-X cards require 5v - and can also have different slot keying...
Also - the extra physical length of the PCI-X card (it will hang outside the standard PCI slot) may interfere with motherboard components - ie heat sinks - as the designer will not have accounted for fitting PCI-X cards as a requirement...
Buy all means give the card a try in a standard PCI slot - it may work.... If not you could always look into a cheap server system - you can HP systems for £300...
I'm looking at expanding my slots and came across this thread. I know it's been inactive for a while but wanted to clarify some things on here for others that may come across it.
PCI-e is not physically compatible with PCI-X or PCI so the OP has a PCI-X card and there's no way to plug it in as suggested-phisically.
Although the OP doesn't have a PCI-X slot, which is what he/she is in need of, keep in mind that not all PCI-X cards will work in a PCI slot and vice-versa. Only cards that indicate they are 32/64 bit will be able to, i.e. Intel Pro 1000 NIC will work in both PCI and PCI-X slots, at least the models I have.
There are riser cards for both PCI-e and PCI-X/PCI that extend the slot so keep that in mind as that is my current problem, I have 1 PCI-X, 1 PCI, and one PCI-e and it is no longer enough. So if you have one PCI-X slot, you can get a dual or tri slot riser card and double or triple your slots. Now it does present a problem, at least for desktops like mine, it means I'll have to seriously modify my desktop cage where the cards get their external connections since it will be going from vertical to horizontal-a lot of work that may or may not be worth it.
There are also PCIx riser cards that are translators, as vineetd posted, that may solve the OP's problem. They translate PCI-e to PCI-X and PCI and there are others that do different translation-but you have the same problem in that the orientation of the cards are now moved 90 degrees-and that may hit other devices in the case. These are called either reverse or forward mode riser cards. Most motherboards have a forward mode built in-it's how you get PCIx slots from the bus embedded on the motherboard that usually go through embedded devices first (which is why generally the first devices detected are always embedded). It's going in reverse that is relatively new. The PEX 8114 and PEX 8111 bridges from PLX, as vineetd posted, are such devices and may help the OP out. I know I'm looking closely at them.
In reality, I'm hoping USB 3.0 and/or other technologies (maybe Infiniband which is 40Gb/s and soon will be 100Gb/s) will make high speed communication devices like external disks, Gig-E and 10Gig network connections more viable so you don't have to have a bus to use them. USB 2.0 is already there but for hard drives and NICs, I find them still too slow.