As of patch v1.12, Blizzard allowed the game to be played without CD since they were offering the digital copy of it from their online store.
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- Patch 1.12
--------------------------------------------------------------------------Downloadable Installer Support
- If all required Diablo 2 '.MPQ' files are installed on the
hard drive, the game will no longer require the CD to play.For users that originally performed a 'Full Installation'
and wish to run without the CD, all '.MPQ' files should
be copied from the Diablo 2 CDs to the Diablo 2 directory.
Most users will only need to copy D2Music.mpq from the
Diablo 2 Play CD and/or D2xMusic.mpq from the Lord of
Destruction CD. Mac users will need to copy these music
files and rename them to 'Diablo II Music' and
'Diablo II Expansion Music' respectively.
So, assuming you've done a "full" install of the Diablo 2 game and that you've installed the expansion pack, you'll also need to copy over D2xMusic.mpq from the expansion CD to your installation folder (by default it should be C:\Program Files\Diablo II).
Something they forgot to mention is that you'll also need D2XVIDEO.MPQ as well. Otherwise you'll get the annoying "Insert Expansion Disc" message when you try to start the game.
It should work straight away once you've got those 2 files in your game folder.