A model is developed for the extrusion in the direction of gravity of a slender fluid cylinder from a die of arbitrary shape. Both gravity and surface tension act to stretch and deform the geometry. The model allows for an arbitrary but prescribed viscosity profile, while the effects of extrudate swell are neglected. The solution is found efficiently through the use of a carefully selected axial Lagrangian coordinate and a transformation to a reduced-time variable. Comparisons between the model and extruded glass microstructured optical fibre preforms show that surface tension has a significant effect on the geometry but the model does not capture all of the behaviour observed in practice. Experimental observations are used in conjunction with the model to argue that some deformation, due neither to surface tension nor gravity, occurs in or near the die exit. Methods are considered to overcome deformation due to surface tension.