The download is in two stages. The broswer downloads the .acsm file, which is just a token for the loan/purchace.
Typically, the browser then automatically launches ADE (Digital Editions), which opens the .acsm file and download the 'real' DRM .epub/.pdf file.
There are cases where ADE2.0 does not properly work with library books; for example if a Sony ereader is plugged in.
See if the more reliable ADE1.7.2 works better.
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There are lots of bugs in ADE2.0 (and 2.0.1). Try replacing ADE2.0 with the older but more reliable v1.7.2.
(You can have them both installed at once if you like.)
Version 1.7.2, it is a little difficult to find, available on Adobe site for Windows and for Mac.
http://helpx.adobe.com/digital-editions/kb/cant-install-digital-editio ns.html
The forum software is sometimes corrupting the link above. There shouldn't be a blank in 'editio ns.html'. The following redirects to the same page: http://tinyurl.com/diged172
Some people have found ADE trying to upgrade automatically.
It appears (not 100% sure) that if you install ADE2.0 as a new install (not as an upgrade) that your 1.7.2 will continue to run.
Probably best to say no if 2.0.x installation asks if you want to migrate your library.