  We present a parametric study of the diffuse magnetic scattering at
(1/2,1/2,1/2) positions in reciprocal space, ascribed to a frozen
antiferromagnetic spin ice state in single crystalline Tb2Ti2O7. Our
high-resolution neutron scattering measurements show that the elastic (-0.02
meV < E < 0.02 meV) (1/2,1/2,1/2) scattering develops strongly below ~275 mK,
and correlates with the opening of a spin gap of ~0.06-0.08 meV over most of
the Brillouin zone. The concomitant low-lying magnetic spin excitations are
weakly dispersive and appear to soften near the (1/2,1/2,1/2) wave vector at 80
mK. The nature of the transition at 275 mK has many characteristics of spin
glass behavior, consistent with ac-susceptibility measurements. The application
of a magnetic field of 0.075 T applied along the [1-10] direction destroys the
(1/2,1/2,1/2) elastic scattering, revealing the fragility of this short-range
ordered ground state. We construct a refined H-T phase diagram for Tb2Ti2O7 and
[1-10] fields which incorporates this frozen spin ice regime and the
antiferromagnetic long-range order previously known to be induced in relatively
large fields. Specific heat measurements on the same crystal reveal a sharp
anomaly at Tc~450 mK and no indication of a transition near ~275 mK. We
conclude that the higher temperature specific heat peak is not related to the
magnetic ordering but is likely a signal of other, nonmagnetic, correlations.
