  This paper concerns the quantisation of a rigid body in the framework of
``covariant quantum mechanics'' on a curved spacetime with absolute time.
  The basic idea is to consider the multi-configuration space, i.e. the
configuration space for $n$ particles, as the $n$-fold product of the
configuration space for one particle. Then we impose a rigid constraint on the
multi-configuration space. The resulting space is then dealt with as a
configuration space of a single abstract `particle'. The same idea is applied
to all geometric and dynamical structures.
  We show that the above configuration space fits into the general framework of
``covariant quantum mechanics''. Hence, the methods of this theory can be
applied to the rigid body.
  Accordingly, we find exactly two inequivalent choices of quantum structures
for the rigid body. Then, we evaluate the quantum energy and momentum operators
and the `rotational part' of their spectra. We provide a new mathematical
interpretation of two-valued wavefunctions on SO(3) in terms of single-valued
sections of a new non-trivial quantum bundle. These results have clear
analogies with spin.
