Physics-augmented neural networks (PANNs) become increasingly popular in constitutive modelling, cf. [1]. Likewise, for the prediction of fracture, the phase-field approach has become a well-established tool. This contribution aims at combining the two approaches, establishing a hybrid model. In doing so, special attention is paid to the split of the free energy into isochoric and dilatational portions. Such a split is essential for adequately modelling fracture of incompressible solids, and, moreover, particularly relevant for phase-field models that incorporate a split of the free-energy into degraded and non-degraded parts. At first, a polyconvex invariant-based PANN that enables the isochoric-volumetric split of the free energy is introduced. Then, the hybrid phase-field model is introduced based on a three-field variational principle, combining the PANN model of the response of the bulk with the classical phase-field approach to fracture. Considering incompressible rubbery polymers, we validate our approach against experimental data, and very good agreement is obtained. In particular, the proposed PANN architecture allows for weaking the incompressibility constraint in fracturing zones, avoiding unphysical phenomena in case of crack growth, see e.g. [2]. [1] L. Linden et al., ‘Neural networks meet hyperelasticity: A guide to enforcing physics’, Journal of the Mechanics and Physics of Solids, 2023. [2] I. Ang et al., ‘Stabilized formulation for phase-field fracture in nearly incompressible hyperelasticity’, International Journal for Numerical Methods in Engineering, 2022.