In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight x(1 − x), x ∈ [0, 1], α, β > −1, the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval [t, 1] is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval (− a, a), a > 0 is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight (1 − x), x ∈ [− 1, 1].