Ark of the Covenant

I just watched a History Channel special on the Ark of the Covenant that was good at showing lots of interesting places, but not so good at discussing the Ark's fate. They spent all their time in Ethiopia, a rather poor bet when your case is all based on legends and local traditions. The chances that Solomon and the Queen of Sheba had a son named Menelik who took the Ark there are especially...remote. Sticking close to the primary sources, it seems certain the Ark was still in Jerusalem during the last days of Judea (Jeremiah 3:16, with support from 2 Chronicles 35:3, though I trust that less). Then the Babylonians destroyed Jerusalem, which seems sufficient to explain the Ark's being lost. Someone familiar with what the Babylonians did with such things can probably speculate from there on what happened to it.