We present results of a lattice computation of the matrix elements of the vector and axial-vector currents which are relevant for the semi-leptonic decays D→K and D→K∗. The computations are performed in the quenched approximation to lattice QCD on a 243×48 lattice at β=6.2, using an O(a)-improved fermionic action. In the limit of zero lepton masses the semi-leptonic decays D→K and D→K∗ are described by four form factors: f+K,V,A1 and A2, which are functions of q2, where qμ is the four-momentum transferred in the process. Our results for these form factors at q2=0 are: $f^+_K(0)=0.67 \er{7}{8}$ , $V(0)=1.01 \err{30}{13}$ , $A_1(0)=0.70 \err{7}{10}$ , $A_2(0)=0.66 \err{10}{15}$ , which are consistent with the most recent experimental world average values. We have also determined the q2 dependence of the form factors, which we find to be reasonably well described by a simple pole-dominance model. Results for other form factors, including those relevant to the decays \dpi and \drho, are also given.