Yes. That should not be a problem. You can leave the installation as is and simply insert the other disc.
Should it not work, backup the folder documents\electronicarts\thesims3\ (that's all the user data like savegames), uninstall only Late Night and reinstall it from the new disc.
If you don't want to buy another disc, you could consider registering the code at Origin and get the download version for LN. It can be used on up to 5 computers. You will need to uninstall LN and then download that version though.
Either way, if you backup that folder, you won't loose any savegames.
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Please accept a solution if your issue is solved. XP is also appreciated.
