  The O(n) loop model on the honeycomb lattice with mixed ordinary and special
boundary conditions is solved exactly by means of the Bethe ansatz. The
calculation of the dominant finite-size corrections to the eigenspectrum yields
the mixed boundary scaling index and the geometric scaling dimensions
describing the universal surface critical behaviour. Exact results follow in
the limit n=0 for the polymer adsorption transition with a mixed adsorbing and
free boundary. These include the new configurational exponent
$\gamma_1=\frac{85}{64}$.
