A high brightness laser-plasma X-ray source has been set-up and is presently available at IFAM. A wide range of diagnostics has been set up to monitor the properties of the X-ray radiation and to control the main parameters including photon energy, flux intensity, and pulse duration. A beam extractor enables access to the X-ray radiation at atmospheric pressure. A simple, easy-to-use projection microscope has been built which is capable of single-shot micron resolution imaging with digital acquisition. Preliminary biomedical experiments show that the X-ray doses available on a single laser shot exposure of our source fully meet the conditions required for an important class of biological experiments based on X-ray induced DNA damage providing an ideal alternative to the long time exposures needed with X-ray tubes.