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. Partial covering by either neutral or partially ionized gas cannot reproduce the spectral shape and, even if its presence is assumed, it does not significantly change the inferred broad-line parameters. By assuming that the inner edge of the accretion disc corresponds to the innermost stable circular orbit of the black hole space–time, the line profile enables us to measure a black hole spin a= 0.6 ± 0.2. However, a non-rotating Schwarzschild space–time is excluded at just above the 3σ level, while a maximally spinning Kerr black hole is excluded at the ∼5σ level. The intermediate spin we measure may indicate a recent merger, or 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. We also report tentative evidence for short time-scale line profile variability. The relatively low statistical significance of the variability (about 98 per cent confidence level) prevents us from drawing any firm conclusions which must be deferred to future observations.