  We have observed the hard X-ray selected Narrow-Line Seyfert 1 galaxy SWIFT
J2127.4+5654 with Suzaku. We report the detection of a broad relativistic iron
emission line from the inner accretion disc. By assuming that the inner edge of
the accretion disc corresponds to the innermost stable circular orbit of the
black hole spacetime, the line profile enables us to measure a black hole spin
a= 0.6\pm0.2. However, a non-rotating Schwarzschild spacetime is excluded at
just above the 3 sigma level, while a maximally spinning Kerr black hole is
excluded at the ~5 sigma level. The intermediate spin we measure may indicate
that accretion-driven black hole growth in this source proceeds through
short-lived episodes with chaotic angular momentum alignment between the disc
and the hole rather than via prolonged accretion. The measured steep emissivity
index (q~5) constrains the irradiating X-ray source to be very centrally
concentrated. Light bending may help focus the primary emission towards the
innermost accretion disc, thus steepening the irradiation profile. On the other
hand, steep profiles can also be reached if magnetic extraction of the hole
rotational energy is at work. If this is the case, the interplay between
accretion (spinning up the black hole) and rotational energy extraction
(spinning it down) forces the hole to reach an equilibrium spin value which,
under standard assumptions, is remarkably consistent with our measurement.
Rotational energy extraction would then be able to simultaneously account for
the intermediate spin and steep emissivity profile we infer from our spectral
analysis without the need to invoke chaotic accretion episodes. (abridged)
