Thermodynamic and ground-state properties of Cu clusters have been studied with the effective-medium theory including a tight-binding description of the one-electron spectrum. Simulated-annealing Monte Carlo calculations have been performed for cluster sizes between 3 and 29 to determine ground-state energies and structures. Finite-temperature ensembles have been generated for a range of temperatures. The magic numbers 8, 18 and 20 are reproduced and remain stable at temperatures close to 1000 K, where the clusters may be regarded as liquid. The coupling between the atomic and electronic degrees of freedom through a Jahn-Teller-like effect is shown to play a key role in understanding the stability of the magic clusters at high temperatures. For the non-magic clusters this Jahn-Teller effect gives rise to large Fermi gaps even at high temperatures and a pronounced stability of the even-sized clusters relative to those with odd sizes. Finite-temperature electronic spectra are calculated. The fluctuations in the atomic positions give rise to a large broadening of the electronic levels, in agreement with experimental observations. Cu₁₃ exhibits a rather sharp melting transition, whereas clusters of other sizes show more complex behaviour.