Let R be a differential domain finitely generated over a differential field F of characteristic 0. Let C be the subfield of differential constants of F. This paper investigates conditions on differential ideals of R that are necessary or sufficient to guarantee that C is also the set of constants of differentiation of the quotient field, E, of R. In particular, when C is algebraically closed and R has a finite number of height one differential prime ideals, there are no new constants in E. An example where F is infinitely generated over C shows the converse is false. If F is finitely generated over C and R is a polynomial ring over F, sufficient conditions on F are given so that no new constants in E does imply only finitely many height one prime differential ideals in R. In particular, F can be (Q) over bar (T) where T is a finite transcendence set.
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