You can download the original 1582 & 1609 Doway-Rhemes Bible in pdf format at: http://www.the-freelance.com/doway-rhemes.html
This is hands-down my absolutely favorite Bible. The modern (post-18th century) "Douay-Rheims" is a significant revision of the original Doway-Rhemes. The Catholic Encyclopedia of 1909 A.D. states:
"Although the Bibles in use at the present day by the Catholics of England and Ireland are popularly styled the Douay Version, they are most improperly so called; they are founded, with more or less alteration, on a series of revisions undertaken by Bishop Challoner in 1749-52 . . .
The changes introduced by him were so considerable that, according to Cardinal Newman, they almost amounted to a new translation. So, also, Cardinal Wiseman wrote, 'To call it any longer the Douay or Rheimish Version is an abuse of terms. It has been altered and modified until scarcely any verse remains as it was originally published.' In nearly every case Challoner's changes took the form of approximating to the Authorized Version [King James]..."
The language is a bit stiff and can be a little difficult to understand at times, and the typography conventions take a little getting used to -- vv instead of w, etc. -- but the footnotes pull no punches and the whole thing is just remarkable, being a word-for-word LITERAL translation of the Vulgate.
Regarding the footnotes, as Matto pointed out:
...the notes are wonderful. They focus on refuting the errors of the protestant sects that were attacking the Church at the time. I would recommend the original Douay Rheims to anyone who asks which bible they should read.