  We report on the first year of Fermi gamma-ray observations of pulsed
high-energy emission from the old PSR J2043+2740. The study of the gamma-ray
efficiency of such old pulsars gives us an insight into the evolution of
pulsars' ability to emit in gammma rays as they age. The gamma-ray lightcurve
of this pulsar above 0.1 GeV is clearly defined by two sharp peaks,
0.353+/-0.035 periods apart. We have combined the gamma-ray profile
characteristics of PSR J2043+2740 with the geometrical properties of the
pulsar's radio emission, derived from radio polarization data, and constrained
the pulsar-beam geometry in the framework of a Two Pole Caustic and an Outer
Gap model. The ranges of magnetic inclination and viewing angle were determined
to be {alpha,zeta}~{52-57,61-68} for the Two Pole Caustic model, and
{alpha,zeta}~{62-73,74-81} and {alpha,zeta}~{72-83,60-75} for the Outer Gap
model. Based on this geometry, we assess possible birth locations for this
pulsar and derive a likely proper motion, sufficiently high to be measurable
with VLBI. At a characteristic age of 1.2 Myr, PSR J2043+2740 is the third
oldest of all discovered, non-recycled, gamma-ray pulsars: it is twice as old
as the next oldest, PSR J0357+32, and younger only than the recently discovered
PSR J1836+5925 and PSR J2055+25, both of which are at least 5 and 10 times less
energetic, respectively.
