We present an extension of general relativity in which an $f(R)$ term `{a} la Palatini is added to the usual metric Einstein-Hilbert Lagrangian. Expressing the theory in a dynamically equivalent scalar-tensor form, we show that it can pass the Solar
System observational tests even if the scalar field is very light or massless. Applications to cosmology and astrophysics, and some exact solutions are discussed.
