The steady motion of an advancing meniscus in a gas‐filled capillary tube involves a delicate balance of capillary, viscous, and intermolecular forces. The limit of small capillary numbers Ca (dimensionless speeds) is analyzed here with a matched asymptotic analysis that links the outer capillary region to the precursor film in front of the meniscus through a lubricating film. The meniscus shape in the outer region is constructed and the apparent dynamic contact angle &THgr; that the meniscus forms with the solid surface is derived as a function of the capillary number, the capillary radius, and the Hamaker’s constant for intermolecular forces, under conditions of weak gas–solid interaction, which lead to fast spreading of the precursor film and weak intermolecular forces relative to viscous forces within the lubricating film. The dependence on intermolecular forces is very weak and the contact angle expression has a tight upper bound tan &THgr;=7.48 Ca1/3for thick films, which is independent of the Hamaker constant. This upper bound is in very good agreement with existing experimental data for wetting fluids in any capillary and for partially wetting fluids in a prewetted capillary. Significant correction to the Ca1/3dependence occurs only at very low Ca, where the intermolecular forces become more important and tan &THgr; diverges slightly from the above asymptotic behavior toward lower values.