  We report the discovery and timing measurements of PSR J1208-6238, a young
and highly magnetized gamma-ray pulsar, with a spin period of 440 ms. The
pulsar was discovered in gamma-ray photon data from the Fermi Large Area
Telescope (LAT) during a blind-search survey of unidentified LAT sources,
running on the distributed volunteer computing system Einstein@Home. No radio
pulsations were detected in dedicated follow-up searches with the Parkes radio
telescope, with a flux density upper limit at 1369 MHz of 30 $\mu$Jy. By timing
this pulsar's gamma-ray pulsations, we measure its braking index over five
years of LAT observations to be $n = 2.598 \pm 0.001 \pm 0.1$, where the first
uncertainty is statistical and the second estimates the bias due to timing
noise. Assuming its braking index has been similar since birth, the pulsar has
an estimated age of around 2,700 yr, making it the youngest pulsar to be found
in a blind search of gamma-ray data and the youngest known radio-quiet
gamma-ray pulsar. Despite its young age the pulsar is not associated with any
known supernova remnant or pulsar wind nebula. The pulsar's inferred dipolar
surface magnetic field strength is $3.8 \times 10^{13}$ G, almost 90% of the
quantum-critical level. We investigate some potential physical causes of the
braking index deviating from the simple dipole model but find that LAT data
covering a longer time interval will be necessary to distinguish between these.
