I did this several times and I'm sure it works, i did it yesterday when my brother needed help with the same problem. It might not be the cleanest method and wasn't listed in the comment, so i just describe what i did exactly:
Step 1: DELETE EVERYTHING!
Yes, you've read it correctly. Right click on the game and press the "Delete local files" button. You need to remove everything at first, otherwise, steam will always continue to download.
Step 2: Terminate steam
Don't just go offline. Terminate it via "exit" in the steam menu. Or kill the process. I prefer the last option since i know it's gone for sure then.
Step 3: Copy the game
Copy the folder with the downloaded game into the
steamapps\common folder. Be sure to use the correct name. In case of GTAV, the folder needs to be names
Grand Theft Auto V. If this isn't the case, steam won't detect it.
Step 4: Install
Restart steam, log in and you will see that GTA (or whatever game you're trying to transfer) is not detected yet. Left click on the game and hit the install button. Steam will now detect existing files (took about 10 - 15 minutes on the pc of my brother who did the same btw) and redownload just a small amount of files (around 200 MB).
After you did all that, you're ready to go. It all worked yesterday when i helped my brother with that problem (it would have taken 6 days to download it)