We report the observation of a Pt layer thickness dependence on the induced interfacial Dzyaloshinskii-Moriya interaction in ultrathin Pt(d_{Pt})/CoFeB films. Taking advantage of the large spin-orbit coupling of the heavy metal, the interfacial Dzyaloshinskii-Moriya interaction is quantified by Brillouin light scattering measurements of the frequency nonreciprocity of spin waves in the ferromagnet. The magnitude of the induced Dzyaloshinskii-Moriya coupling is found to saturate to a value of 0.45 mJ/m^{2} for Pt thicknesses larger than ∼2 nm. The experimental results are explained by analytical calculations based on the three-site indirect exchange mechanism that predicts a Dzyaloshinskii-Moriya interaction at the interface between a ferromagnetic thin layer and a heavy metal. Our findings open up a way to control and optimize chiral effects in ferromagnetic thin films through the thickness of the heavy-metal layer.