  We use a method, first developed for the Riemann zeta-function by Masser in
["Rational values of the Riemann zeta function", Journ. Num. Th. 131 (2011),
2037-2046], to prove a new zero estimate for polynomials in z and 1/Gamma(z).
This allows us to prove that, for all n>=2, there exists an absolute effective
positive constant C(n) such that, for all D>=3, there are at most
C(n)log^2(D)/loglog(D) rational numbers z in [n-1,n] with denominator at most D
and such that Gamma(z) is also rational with denominator at most D.
