In cardiac MRI, different diagnostic parameters are obtained in separate scans, leading to long examination times. In this work, we present an iterative model-based reconstruction approach for continuously acquired data, which provides native T1 maps and functional cine images within a single breath hold. The continuous acquisition allows for T1 reconstruction for different cardiac phases. Evaluation in a phantom demonstrated accurate T1 values (R²>0.99) and insensitivity to heart rates, with T1 variations of less than 5% (50 to 90 bpm). In three healthy volunteers T1 maps were assessed for diastole and systole and cine images had a consistent dark-blood contrast.