The cave is B?ri in the Leitahraun south of Reyjlav?k. The cave was discovered by two friends of mine, Gu?mundur-Brynyar ??rsteinsson (aka Bibi) and Bj?rn Hr?arsson in early 2005. I was lucky enough to be part of the small team that surveyed the cave in June 2005. The cave is almost exactly 1km long and has some very large passages and nice features, and well worth a visit if you are in the area (though the entrance dig is somewhat "interesting"). I've added a few photos of the cave just in case anyone is interested...
I'm not sure the commentary is correct about the pitch at the end of the cave being the deepest in a lava tube, I can think of a few others of at least similar depth in the Canaries and Jeju Island at the very least.