Despite recent successes in the synthesis of boron nanotubes (BNTs), the atomic arrangement of their walls has not yet been determined and many questions about their basic properties do remain. Here, we unveil the dynamical stability of BNTs by means of first-principles molecular dynamics simulations. We find that free-standing, single-wall BNTs with diameters larger than 0.6 nm are thermally stable at the experimentally reported synthesis temperature of 870 degrees Celsius and higher. The walls of thermally stable BNTs are found to have mixed triangular-hexagonal morphologies. The latter could develop into a new structural paradigm in boron chemistry.