  Fast Radio Bursts are bright, unresolved, non-repeating, broadband,
millisecond flashes, found primarily at high Galactic latitudes, with
dispersion measures much larger than expected for a Galactic source. The
inferred all-sky burst rate is comparable to the core-collapse supernova rate
out to redshift 0.5. If the observed dispersion measures are assumed to be
dominated by the intergalactic medium, the sources are at cosmological
distances with redshifts of 0.2 to 1. These parameters are consistent with a
wide range of source models. One fast radio burst showed circular polarization
[21(7)%] of the radio emission, but no linear polarization was detected, and
hence no Faraday rotation measure could be determined. Here we report the
examination of archival data revealing Faraday rotation in a newly detected
burst - FRB 110523. It has radio flux at least 0.6 Jy and dispersion measure
623.30(5) pc cm$^{-3}$. Using Galactic contribution 45 pc cm$^{-3}$ and a model
of intergalactic electron density, we place the source at a maximum redshift of
0.5. The burst has rotation measure -186.1(1.4) rad m$^{-2}$, much higher than
expected for this line of sight through the Milky Way and the intergalactic
medium, indicating magnetization in the vicinity of the source itself or within
a host galaxy. The pulse was scattered by two distinct plasma screens during
propagation, which requires either a dense nebula associated with the source or
a location within the central region of its host galaxy. Keeping in mind that
there may be more than one type of fast radio burst source, the detection in
this instance of source-local magnetization and scattering favours models
involving young stellar populations such as magnetars over models involving the
mergers of older neutron stars, which are more likely to be located in low
density regions of the host galaxy.
