  We study asymptotic behavior for determinants of $n\times n$ Toeplitz
matrices corresponding to symbols with two Fisher-Hartwig singularities at the
distance $2t\ge0$ from each other on the unit circle. We obtain large $n$
asymptotics which are uniform for $0<t<t_0$ where $t_0$ is fixed. They describe
the transition as $t\to 0$ between the asymptotic regimes of 2 singularities
and 1 singularity. The asymptotics involve a particular solution to the
Painlev\'e V equation. We obtain small and large argument expansions of this
solution. As applications of our results we prove a conjecture of Dyson on the
largest occupation number in the ground state of a one-dimensional Bose gas,
and a conjecture of Fyodorov and Keating on the second moment of powers of the
characteristic polynomials of random matrices.
