  A Brill-Noether locus is a subvariety of M_g consisting of curves having
certain linear series g^r_d. We study the relative position of Brill-Noether
loci with respect to the gonality stratification of M_g. We construct smooth
curves in P^r of given degree and genus and having the `expected' gonality. As
an application we give a new proof of our result about the Kodaira dimension of
the moduli space of curves of genus 23.
