The long-time dynamics of the critical contact process which is brought suddenly out of an uncorrelated initial state undergoes ageing in close analogy with quenched magnetic systems. In particular, we show through Monte Carlo simulations in one and two dimensions and through mean-field theory that time-translation invariance is broken and that dynamical scaling holds. We find that the autocorrelation and autoresponse exponents lambda_Gamma and lambda_R are equal but, in contrast to systems relaxing to equilibrium, the ageing exponents a and b are distinct. A recent proposal to define a non-equilibrium temperature through the short-time limit of the fluctuation-dissipation ratio is therefore not applicable.

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