Motivated by a recent paper by the Potsdam numerical relativity group, we have constructed a new numerical code for hydrodynamic simulation of axisymmetric systems in full general relativity. In this code, we solve the Einstein field equation using Cartesian coordinates with appropriate boundary conditions. On the other hand, the hydrodynamic equations are solved in cylindrical coordinates. Using this code, we perform simulations to study axisymmetric collapse Of rotating stars, which thereby become black holes or new compact stars, in full general relativity. To investigate the effects of rotation on the criterion for prompt collapse to black holes, we first adopt a polytropic equation of state, P = K rho(Gamma), where P, rho, and K are the pressure, rest mass density, and polytropic constant, with Gamma = 2. In this case, the collapse is adiabatic (i.e., no change in entropy), and we can focus on the bare effect of rotation. As the initial conditions, we prepare rigidly and differentially rotating stars in equilibrium and then decrease the pressure to induce collapse. In this paper, we consider cases in which q equivalent to J/M-g(2) < 1, where J and M-g are the angular momentum and the gravitational mass. It is found that the criterion of black hole formation is strongly dependent on the angular momentum parameter q. For q < 0.5, the criterion is not strongly sensitive to q; more precisely, if the rest mass is slightly larger than the maximum allowed value of spherical stars, a black hole is formed. However, for q less than or similar to 1, it changes significantly: For q similar or equal to 0.9, the maximum allowed rest mass becomes similar to 70-80% larger than that for spherical stars. These findings depend only weakly on the rotational profiles given initially. We then report the results for simulations employing a Gamma-law equation of state P = (Gamma - 1)rho epsilon, where epsilon is the specific internal energy, to study effects of shock heating. We find that the effects of shock heating are particularly important for preventing prompt collapse to black holes in the case of large q [i.e., q = 0(1)].