  Exact results for conformational statistics of compact polymers are derived
from the two-flavour fully packed loop model on the square lattice. This loop
model exhibits a two-dimensional manifold of critical fixed points each one
characterised by an infinite set of geometrical scaling dimensions. We
calculate these dimensions exactly by mapping the loop model to an interface
model whose scaling limit is described by a Liouville field theory. The
formulae for the central charge and the first few scaling dimensions are
compared to numerical transfer matrix results and excellent agreement is found.
Compact polymers are identified with a particular point in the phase diagram of
the loop model, and the non-mean field value of the conformational exponent
\gamma = 117/112 is calculated for the first time. Interacting compact polymers
are described by a line of fixed points along which \gamma varies continuously.
