  We measure exclusive $\pi^+\pi^-$ production in proton-antiproton collisions
at center-of-mass energies $\sqrt{s}$ = 0.9 and 1.96 TeV in the Collider
Detector at Fermilab. We select events with two oppositely charged particles,
assumed to be pions, with pseudorapidity $|\eta| < 1.3$ and with no other
particles detected in $|\eta| < 5.9$. We require the $\pi^+\pi^-$ system to
have rapidity $|y|<$ 1.0. The production mechanism of these events is expected
to be dominated by double pomeron exchange, which constrains the quantum
numbers of the central state. The data are potentially valuable for isoscalar
meson spectroscopy and for understanding the pomeron in a region of transition
between nonperturbative and perturbative quantum chromodynamics. The data
extend up to dipion mass $M(\pi^+\pi^-)$ = 5000 MeV/$c^2$ and show resonance
structures attributed to $f_0$ and $f_2(1270)$ mesons. From the $\pi^+\pi^-$
and $K^+K^-$ spectra, we place upper limits on exclusive $\chi_{c0}(3415)$
production.
