  The transition to turbulence in a precessing cylindrical vessel is
experimentally investigated. Our measurements are performed for a {
nearly-resonant} configuration with an initially laminar flow dominated by an
inertial mode with azimuthal wave number $m=1$ superimposed on a solid body
rotation. By increasing the precession ratio, we observe a transition from the
laminar to a non-linear regime, which then breakdowns to turbulence for larger
precession ratio. Our measurements show that the transition to turbulence is
subcritical, with a discontinuity of the wall-pressure and the power
consumption at the threshold $\epsilon_{LT}$. The turbulence is self-sustained
below this threshold, describing a bifurcation diagram with a hysteresis. In
this range of the control parameters, the turbulent flows can suddenly collapse
after a finite duration, leading to a definitive relaminarization of the flow.
The average lifetime $\langle \tau \rangle$ of the turbulence increases rapidly
when $\epsilon$ tends to $\epsilon_{LT}$.
