The axisymmetric capillary pinch-off of a viscous fluid thread of viscosity $\lambda\mu$ and surface tension $\gamma$ immersed in a surrounding fluid of viscosity $\mu$ is studied. Similarity variables are introduced (with lengthscales decreasing like $\tau$, the time to pinch-off, in a rapidly translating frame) and the self-similar shape is determined directly by a combination of modified Newton iteration and a standard boundary-integral method. A large range of viscosity ratios is studied ($0.002\,{\leq}\,\lambda\,{\leq}\, 500$) and asymmetric profiles are observed for all $\lambda$, with conical shapes far from the pinching point, in agreement with previous time-dependent studies. The stability of the steady solutions is investigated and oscillatory instability is found for $\lambda \ge 32$. For $\lambda\,{\ll}\, 1$ an asymptotic scaling of $\lambda^{1/2}$ is suggested for the slopes of the far-field conical shapes. These compare well with the quantitative predictions of a one-dimensional theory based on Taylor's (1964) analysis of a slender bubble.