Wallace Shawn did the translation.
The 1954 Marc Blitzstein translation is the most well-known. He was an extremely adroit song lyricist and those versions have become part of the American musical theatre songbook. That version also propelled Lenya into a kind of superstardom-- well-deserved indeed. The question is, then why should Threepenny get perennially retranslated? The answer is that it is Brecht and Hauptman's text that seems to have suffered in this adaptation, done at a time when the political climate was not exactly conducive to frankness on the stage.
I am glad that you are familiar with a few versions. The Brecht Foundation, which oversees all things textual regarding his work, tries to hold fast to Brecht's belief that all theatre is a living, breathing and ever-changing thing, and gives the adaptors and translators a fair amount of leeway. The Weill Foundation, trying to preserve the score as a piece of classical theatrical music, is naturally not so adaptable... even down to having to approve the number of musicians playing the score and whether or not the songs may be transposed to fit 21st Century actors' voices. The two foundations are, as a result, often at odds about a production. Certainly a very interesting situation.
And though we will have no soundtrack, we will be documented by the Performing Arts Library at Lincoln Center, so perhaps one day you might go up there and see the show for yourself.