  The one-texture universe, introduced by Davis in 1987, is a homogeneous
mapping of a scalar field with an $S^3$ vacuum into a closed universe. It has
long been known to mathematicians that such solutions, although static, are
unstable. We show by explicit construction that there is only one unstable
mode, corresponding to collapse of the texture towards a single point, in the
case where gravitational backreaction is neglected. We discuss the instability
timescale in both static and expanding space-times; in the latter case it is of
order of the present age of the universe, suggesting that, though unstable, the
one-texture universe could survive to the present. The cosmic microwave
background constrains the initial magnitude of this unstable perturbation to be
less than of order 10^{-3}.
