Thanks @srowen! I couldn't have said it in a better way. Here is another source:
http://web.engr.oregonstate.edu/~tgd/classes/534/slides/part13.pdf, slide 11.

In case all labels are positive or negative there is no real point in doing any classification. I'd be fine redefining the value but personally I think 0 makes the most sense:

Indeed by randomly guessing you can always achieve 0.5, but maybe the model is actually wrong in all labels. Same would hold for using 1.0: this gives the impression the model is perfect while there is no way to test this. Setting it to 0.0 at least signals something is wrong. 

But again, I'd be fine with whatever.