When the genomic integrity of a cell is challenged, its fate is determined in part by signals conveyed by the p53, tumour suppressor protein. It was observed recently that such signals are not simple gradations of p53 concentration, but rather a counter-intuitive limit-cycle behaviour. Based on a careful mathematical interpretation of the experimental body of knowledge, we propose a model for the p53 signalling network and characterise the p53 stability and oscillatory dynamics. In our model, ATM, a protein that senses DNA damage, activates p53 by phosphorylation. In its active state, p53 has a decreased degradation rate and an enhanced transactivation of Mdm2, a gene whose protein product Mdm2 tags p53 for degradation. Thus the p53-Mdm2 system forms a negative feedback loop. However, the feedback in this loop is delayed, as the pool of Mdm2 molecules being induced by p53 at a given time will mark for degradation the pool of p53 molecules at some later time, after the Mdm2 molecules have been transcribed, exported out of the nucleus, translated and transported back into the nucleus. The analysis of our model demonstrates how this time lag combines with the ATM-controlled feedback strength and effective dampening of the negative feedback loop to produce limit-cycle oscillations. The picture that emerges is that ATM, once activated by DNA damage, makes the p53-Mdm2 oscillator undergo a supercritical Hopf bifurcation. This approach yields an improved understanding of the global dynamics and bifurcation structure of our time-delayed, negative feedback model and allows for predictions of the behaviour of the p53 system under different perturbations.