  We generalize the main result of arXiv:1206.6631 [math.NT] to all totally
real fields. In other words, for $p>2$ prime, we prove (under a mild
Taylor-Wiles hypothesis) that if a modular representation is unramified and
$p$-distinguished at all places above $p$, then it arises from a mod $p$
Hilbert modular form of parallel weight one. This (mostly) resolves the weight
one part of Serre's conjecture for totally real fields.
