  Let R be a differential domain finitely generated over a differential field,
F, with field of constants, C, of characteristic 0. Let E be the quotient field
of R. The paper investigates necessary and sufficient conditions on R's
differential ideals for the constants of differentiation of E to also be C (no
new constants). It was known that no proper nonzero differential ideals
guarantees no new constants. The paper shows that when C is algebraically
closed that if R has only finitely many height one prime differential ideals
that E will have no new constants. An example with F infinitely generated over
C shows the converse is false. The paper gives conditions on C and F so that
when F is finitely generated over C and R is a polynomial ring over F that no
new constants in E implies that R has only fintely many height one prime
differential ideals.
