  Spontaneous motion due to symmetry breaking has been theoretically predicted
for both active droplets and isotropically active particles in an unbounded
fluid domain, provided their intrinsic P\'eclet number $Pe$ exceeds a critical
value. However, due to their inherently small $Pe$, this phenomenon has yet to
be experimentally observed for active particles. In this paper, we
theoretically demonstrate that spontaneous motion for an active spherical
particle closely fitting in a cylindrical channel is possible at arbitrarily
small $Pe$. Scaling arguments in the limit where the dimensionless clearance
$\epsilon\ll1$ reveal that when $Pe=O(\epsilon^{1/2})$, the confined particle
reaches speeds comparable to those achieved in an unbounded fluid at moderate
(supercritical) $Pe$ values. We use matched asymptotic expansions in that
distinguished limit, where the fluid domain decomposes into several asymptotic
regions: a gap region, where the lubrication approximation applies;
particle-scale regions, where the concentration is uniform; and far-field
regions, where solute transport is one-dimensional. We derive an asymptotic
formula for the particle speed, which is a monotonically decreasing function of
$\overline{Pe}=Pe/\epsilon^{1/2}$ and approaches a finite limit as
$\overline{Pe}\searrow0$. Our results could pave the way for experimental
realisations of symmetry-breaking spontaneous motion in active particles.
