  The distance distribution between complementary base pairs of the two strands
of a DNA molecule is studied near the melting transition. Scaling arguments are
presented for a generalized Poland-Scheraga type model which includes
self-avoiding interactions. At the transition temperature and for a large
distance r the distribution decays as 1/r^\kappa with \kappa=1+(c-2)/ \nu. Here
\nu is the self-avoiding walk correlation length exponent and c is the exponent
associated with the entropy of an open loop in the chain. Results for the
distribution function just below the melting point are also presented.
Numerical simulations which fully take into account the self-avoiding
interactions are in good agreement with the scaling approach.
