Fluctuations in the cosmic microwave background (CMB) contain information which has been pivotal in establishing the current cosmological model. These data can also be used to test well-motivated additions to this model, such as cosmic textures. Textures are a type of topological defect that can be produced during a cosmological phase transition in the early Universe, and which leave characteristic hot and cold spots in the CMB. We apply Bayesian methods to carry out a rigorous test of the texture hypothesis, using full-sky data from the Wilkinson Microwave Anisotropy Probe. We conclude that current data do not warrant augmenting the standard cosmological model with textures. We rule out at 95% confidence models that predict more than 6 detectable cosmic textures on the full sky.