We study the possible weights of an irreducible two-dimensional mod p
representation of the absolute Galois group of F which is modular in the sense
of that it comes from an automorphic form on a definite quaternion algebra with
centre F which is ramified at all places dividing p, where F is a totally real
field. In most cases we determine the precise list of possible weights; in the
remaining cases we determine the possible weights up to a short and explicit
list of exceptions.