We propose a modified gravity theory that propagates only two local gravitational degrees of freedom and that does not have an Einstein frame. According to the classification in JCAP 01 (2019) 017 [arXiv:1810.01047 [gr-qc]], this is a type-II minimally modified gravity theory. The theory is characterized by the gravitational constant $G_{rm N}$ and a function $V(phi)$ of a non-dynamical auxiliary field $phi$ that plays the role of dark energy. Once one fixes a homogeneous and isotropic cosmological background, the form of $V(phi)$ is determined and the theory no longer possesses a free parameter or a free function, besides $G_{rm N}$. For $V(phi) = 0$ the theory reduces to general relativity (GR) with $G_N$ being the Newtons constant and $V=const.$ being the cosmological constant. For $V(phi) e 0$, it is shown that gravity behaves differently from GR but that GR with $G_{rm N}$ being the Newtons constant is recovered for weak gravity at distance and time scales sufficiently shorter than the scale associated with $V(phi)$. Therefore this theory provides the simplest framework of cosmology in which deviations from GR can be tested by observational data.
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