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The recently discovered superconductor UTe_{2}, with a superconducting transition temperature T_{c} between 1.5 and 2 K, is attracting much attention due to strong suspicion of spin-triplet and topological superconductivity. Its properties under magnetic field are also remarkable, with field-reinforced (H∥b) and field-induced [H in the (b,c) plane] superconducting phases. Here, we report the first complete thermodynamic determination of the phase diagram for fields applied along the three crystallographic directions. For field along the easy a axis, we uncover a strong negative curvature of the upper critical field very close to T_{c}, revealing a strong suppression of the pairing strength at low magnetic fields. By contrast, measurements performed up to 36 T along the hard magnetization b axis confirm a bulk field-reinforced superconducting phase. Most of all, they also reveal the existence of a phase transition line within the superconducting phase. Drastic differences occur between the low-field and high-field phases pointing to different pairing mechanisms. Detailed analysis suggests a possible transition between a low-field spin-triplet to high-field spin-singlet state, a unique case among superconductors.