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 pulsars 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 pulsars 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.
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