Respiratory and cardiac motion leads to image degradation in positron emission tomography (PET) studies of the human heart. In this paper we present a novel approach to motion correction based on dual gating and mass-preserving hyperelastic image registration. Thereby, we account for intensity modulations caused by the highly non-rigid cardiac motion. This leads to accurate and realistic motion estimates which are quantitatively validated on software phantom data and carried over to clinically relevant data using a hardware phantom. For patient data, the proposed method is first evaluated in a high statistic (20 minutes scans) dual gating study of 21 patients. It is shown that the proposed approach properly corrects PET images for dual - cardiac as well as respiratory - motion. In a second study the list mode data of the same patients is cropped to a scan time reasonable for clinical practice (3 minutes). This low statistic study not only shows the clinical applicability of our method but also demonstrates its robustness against noise obtained by hyperelastic regularization.