The recording of Das Buch Mormon appears to have been digitized from cassette tapes produced by the Church, based on information from the site's home page. Recordings were most likely produced from the 1980 edition of the German Book of Mormon – the edition that preceded the current 2003 edition – which explains why the recordings are no longer published by the Church.
The Book of Mormon Project has a PDF that appears to have text from the 1980 edition of the Book of Mormon – based on the copyright date (1985). I don't speak German, so you'll have to let us know if it's the correct edition. :) I was also able to find several used versions of the book in print, for sale online.
From what I can tell, there have been 4 actual translations of the Book of Mormon into German over the years, each of which has had several printings or editions: 1852, 1959, 1980, and 2003. I found some good historical information in the Journal of Book of Mormon Studies, Dialogue: A Journal of Mormon Thought, and Deseret News. I expect there will be a new edition in the next year or two that includes updates from the 2013 edition of the English scriptures, as similar updates have been published in other languages – most recently French, Italian, and Dutch. The current edition of the German Book of Mormon can be found on LDS.org.
If you are interested in looking at the first translation of the German Book of Mormon (1852), which is in the public domain, you can find scans at Archive.org and on Google Books (links below). Good luck parsing the old German Fraktur!