The measured properties of the particle detected six years ago at the LHC at CERN indicate that it is compatible with the Higgs boson predicted by the Standard Model. However, the theoretical and experimental uncertainties allow associations with extended models. Therefore it is of essential importance to investigate the properties of this particle in more detail. The determination of the Higgs potential is crucial to test whether this particle causes electroweak symmetry breaking. The self-coupling strength has to be determined to measure the Higgs potential. This can be achieved in a first step by measuring the trilinear self-coupling in Higgs pair production. At the LHC, the dominant process of Higgs pair production is the loop induced gluon fusion. Therefore the main goal of this thesis is the calculation of the next-to-leading order (NLO) QCD corrections considering the complete top-quark mass dependence in the framework of the Standard Model. The relevant two-loop integrals cannot be calculated analytically with the currently known methods. Instead a numerical integration is required. The main challenge is the extraction of the ultraviolet, the infrared and the collinear divergences from the amplitudes. For this purpose, a modified end-point subtraction has been developed for the extraction of the IR-singularities. Publicly available programs have been used for the real corrections. The differential cross section has been obtained as a distribution in the invariant Higgs pair mass. It shows that the main contributions to the cross section emerge from the invariant Higgs pair masses between 300 and 800 GeV and that the heavy-top limit is a reasonable approximation for invariant Higgs pair masses only up to about 600 GeV. Further, it can be observed that for an invariant Higgs pair mass up to 400-600 GeV the NLO QCD corrections can be reasonably approximated by the K-factor of the triangular contributions alone. The obtained hadronic cross section implies a negative contribution of about −15% from NLO mass effects compared to the previous known heavy-top limit results involving the full leading-order mass dependence.