The subject of this paper is a finite difference scheme for the Hamilton - Jacobi equation \[ \phi_t+H(\nabla_x\phi)=0, \] where \(H\) is the Hamiltonian and \(x=(x_1,x_2,\cdots,x_d)\). The proposed scheme is of central type i.e. values of the function \(\phi\) and its space derivatives are computed (in the case of one space dimension), in the point \(x_{j+{1\over 2}}\), not belonging to the space grid. Also the forward time step is realized by approximation of the integral in the formula \[ \phi(x_{j+{1\over 2}},t^{n+1})=\phi(x_{j+{1\over 2}},t^n)- \int_{t^n}^{t_{n+1}}H(\phi_x(x_{j+{1\over 2}},t)) dt \] with help of the value of \(H\) at the central point \((x_{j+{1\over 2}},t^{n+{1\over 2}})\). In order to compute all the necessary values, interpolation by a polynomial of the degree 2 with delimiters is used. The maximal speed of propagation is estimated on each step and this information is used for final approximation of the value of \(\phi\) at the time level \(t^{n+1}\). Since the proposed scheme is central, there is no problems with ``upwinding''. The used delimiter prevents against parasite oscillations. First and second order, as well as one and multidimensional space versions of the scheme are discussed. The paper contains numerical examples.