  Accretion discs are generally warped. If a warp in a disc is too large, the
disc can `break' apart into two or more distinct planes, with only tenuous
connections between them. Further if an initially planar disc is subject to a
strong differential precession, then it can be torn apart into discrete annuli
that precess effectively independently. In previous investigations,
torque-balance formulae have been used to predict where and when the disc
breaks into distinct parts. In this work, focusing on discs with Keplerian
rotation and where the shearing motions driving the radial communication of the
warp are damped locally by turbulence (the `diffusive' regime), we investigate
the stability of warped discs to determine the precise criterion for an
isolated warped disc to break. We find and solve the dispersion relation, which
in general yields three roots. We provide a comprehensive analysis of this
viscous-warp instability and the emergent growth rates and their dependence on
disc parameters. The physics of the instability can be understood as a
combination of (1) a term which would generally encapsulate the classical
Lightman-Eardley instability in planar discs (given by
$\partial(\nu\Sigma)/\partial\Sigma < 0$) but is here modified by the warp to
include $\partial(\nu_1|\psi|)/\partial|\psi| < 0$ and (2) a similar condition
acting on the diffusion of the warp amplitude given in simplified form by
$\partial(\nu_2|\psi|)/\partial|\psi| < 0$. We discuss our findings in the
context of discs with an imposed precession, and comment on the implications
for different astrophysical systems.
