In section I.6 of Algebraic Geometry, Hartshorne establishes a that every curve is birationally equivalent to a nonsingular projective curve. To do this, he defines for any given curve $C$ with function field $K$ an abstract nonsingular curve $C_K$ whose points are just DVR's of $K$ with an appropriate topology. It can be shown that $C_K$ is isomorphic to a nonsingular projective curve.
Later on in ex II.3.8, Hartshorne defines the normalization of a scheme, which in the case of curves will be a nonsingular curve that is birational to the original curve.
In general, do these two constructions give the same curve? They give birational curves by II.6.12, but I would like to say that they are actually the same. If this is not true, can we at least say that the normalization has an open immersion into $C_K$? I have a hunch that the latter statement should be true since the points of a nonsingular curve just correspond to DVR's of $K.$
Thanks ahead of time.