  We propose two new versions of ghost-free generalized $F(R)$ gravity with
Lagrange multiplier constraint. The first version of such theory for a
particular degenerate choice of the Lagrange multiplier, corresponds to mimetic
$F(R)$ gravity. The second version of such theory is just the Jordan frame
description of mimetic gravity with potential. As we demonstrate, it is
possible to realize several cosmological scenarios in such theory. In
particulary, de Sitter solutions may also be found.
