  Quantum gravity in a finite region of spacetime is conjectured to be dual to
a conformal field theory deformed by the irrelevant operator $T \overline{T}$.
We test this conjecture with entanglement entropy, which is sensitive to
ultraviolet physics on the boundary while also probing the bulk geometry. We
find that the entanglement entropy for an entangling surface consisting of two
antipodal points on a sphere is finite and precisely matches the Ryu-Takayanagi
formula applied to a finite region consistent with the conjecture of McGough,
Mezei and Verlinde. We also consider a one-parameter family of conical
entropies, which are finite and verify a conjecture due to Dong. Since
ultraviolet divergences are local, we conclude that the $T \overline{T}$
deformation acts as an ultraviolet cutoff on the entanglement entropy. Our
results support the conjecture that the $T \overline{T}$-deformed CFT is the
holographic dual of a finite region of spacetime.
