The vocal begins at around 0:20, and it seems to have some kind of saturation that I wasn't able to figure out. Or would that be a consequence of the delay? I tried some tape and tube saturations, but not shure yet. I've heard that effect also on John Newman's voice, and found it so cool.
***Edit : Now that i read your question again, you are actually not asking about ambience but you ask about the distort effect. Well this could be some hi end distortion box, or some vintage tube preamp, but if you have a small guitar amplifier you can come pretty close running vocals through it and distorting a bit, cause not only it distorts but it gives that hi freq cut too.
This effect is mostly done with guitar amps, many people use small vintage hi end guitar amps like 60's the smaller the size of the speaker cone the better.