Modz already did this solo (on day 2 of PTR iirc) but without using the ring. I think this was the VOD but it's only available to subscribers so I can't be sure: http://www.twitch.tv/datmodz/b/542387559
Frostburn Gauntlets are bugged so it procs off cold elemental dmg on weapons, not on chilled mobs, which makes it easy to freeze mobs (or you have the barb do it with cold leaps). You clear the rift by applying EP to several mobs and then waiting for rimeheart proc to instagib one to set off EP killing the rest. Once you get to the RG you teleport out, swap your weapon to The Furnace, and use its procs to take off 6-8% (boosted by elite dmg but not fire) of the RG's current health at a time. Bul-Kathos's Wedding Band drains 0.4% of the enemies current health and heals you for 0.4% of your max health so it's like a smaller more consistent furnace proc if you don't have a furnace but I'm not sure it's worth using both since you only need a few procs to get it very low.