Cannot append segment blah blah blah: The segment has a different set of streams.

After various experimentations, and researching on the Internet, I am pleased to say that I have come up with a tried-and-tested solution. Here are the following steps, and let's call the two videos "Video X" and "Video Y". I know, some of you guys are thinking "Hey, why not 'Video A' and 'Video B', or 'Video 1' and 'Video 2'?" Well, because I am in an X and Y mood, so as far as I am concerned, it's Video X and Video Y. Thank you very much.

1. Fire up VirtualDubMod.
2. In the menu bar, click File, click Open video file, look for Video X and press the Open button.
3. In the menu bar, click Video, and make sure Direct Stream Copy is selected.
4. In the menu bar again, click Streams followed by Stream List. Right-click on the video entitled "Source: Input AVI file", and make sure Direct Stream Copy is selected too.
5. In the menu bar yet again, click Video, then Frame Rate, and select No change.
6. Then click File followed by Save As, assign your desired video output a name, and then click the Save button.
7. Repeat for Video Y.
7. Once both videos have been processed, they should join fine (as long as BOTH share the same frame rate!).