Unfortunately, no.
The root of the problem is that any Install of Windows creates a customized version on your HDD. During Install, the process looks around at all the hardware devices in the machine and ensures that the correct software drivers for all of them are installed. "Devices" is much more than added video cards or a modem in a PCI slot, etc. "Devices" includes mobo chips for control of memory, hard drive controllers, the USB bus system, audio chip, and a whole bunch of other stuff. The net result is that your old HDD contains all the drivers for all those devices in your old machine, but is missing most of the drivers for the devices in a new mobo. Most of the time this means it could not boot at all on the new mobo, or if it did it would have a lot of failures.
As RealBeast said, there is a way to fix this problem that works IF the new mobo is very similar to the old one so that a repair process has a decent chance to correct the mismatches. But for mobos quite different, especially if there is a very different CPU and GPU system, the differences are too great.
The real way to address this problem is to wipe the old HDD clean and completely re-Install Windows. But that means three important things are required:
1. BEFORE you start you must do a complete backup of the entire old HDD, because the process will destroy all your old data on it.
2. You need a Windows Install CD. In your case, having had a pre-built unit, you may not have that. You may have only a Repair CD able to repair corrupted files on the existing Windows version installed.
3. You need ALL the Install CD's for ALL your old software. By Installing Windows fresh, you will create a new set of Registry files that knows NOTHING about your old software. Only by re-Installing each software package can you get the Registry files updated to use them again.
AFTER all that is done, you can use your backup of your data to copy it back to the renewed HDD.
Unfortunately, the straightforward way costs money. Your best path is to buy a new HDD, and MAYBE a new Windows - the latter depends whether you have a Windows Install CD already. After all, if your machine is "very old", maybe the old HDD is going to fail in a few years. If you go this route, the old HDD IS your backup of old data. AFTER you have built the new machine with a new HDD and Installed Windows on it, you Install all your software. THEN you connect the old HDD to the new machine and copy data from it to the new HDD. When that is complete, disconnect and store the old HDD, just in case you find you missed something and need to get it from that "backup".
Eventually you will decide you don't need that old set of backup files, and you could decide to re-install that old HDD in the new machine, wipe it clean, and use it as a second drive for data storage.That is, as long as it appears to be reliable.