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 that would generally encapsulate the classical Lightman-Eardley instability in planar discs (given by partial derivative(nu Sigma)/partial derivative Sigma < 0) but is heremodified by the warp to include partial derivative(nu(1)vertical bar psi vertical bar)/partial derivative vertical bar psi vertical bar < 0, and (2) a similar condition acting on the diffusion of the warp amplitude given in simplified form by partial derivative(nu(2)vertical bar psi vertical bar)/partial derivative vertical bar psi vertical bar < 0. We discuss our findings in the context of discs with an imposed precession, and comment on the implications for different astrophysical systems.