  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.
