We present the results of an experiment to search for trapped antihydrogen atoms with the ALPHA antihydrogen trap at the CERN Antiproton Decelerator. Sensitive diagnostics of the temperatures, sizes, and densities of the trapped antiproton and positron plasmas have been developed, which in turn permitted development of techniques to precisely and reproducibly control the initial experimental parameters. The use of a position-sensitive annihilation vertex detector, together with the capability of controllably quenching the superconducting magnetic minimum trap, enabled us to carry out a high-sensitivity and low-background search for trapped synthesised antihydrogen atoms. We aim to identify the annihilations of antihydrogen atoms held for at least 130 ms in the trap before being released over ~30 ms. After a three-week experimental run in 2009 involving mixing of 107 antiprotons with 1.3ï¿œ109 positrons to produce 6ï¿œ105 antihydrogen atoms, we have identified six antiproton annihilation events that are consistent with the release of trapped antihydrogen. The cosmic ray background, estimated to contribute 0.14 counts, is incompatible with this observation at a significance of 5.6 sigma. Extensive simulations predict that an alternative source of annihilations, the escape of mirror-trapped antiprotons, is highly unlikely, though this possibility has not yet been ruled out experimentally.