The ballistic transmission of the Dirac ultrarelativistic quasielectrons in graphene structures with the rectangular potential barrier is considered, and both the single and the double-barrier structures are analyzed. Within the framework of the continuum model, the transmission coefficient of quasielectrons T is calculated depending on the parameters of the problem. It is believed that there is an electrostatic barrier, as well as the Fermi velocity barrier, due to the fact that this quantity may acquire different values in the barrier and out-of-barrier regions (υF2 and υF1, respectively) of the considered structures. It is shown that the effect of the supertunneling manifests itself in these structures which consists in the fact that under certain conditions the transmission through the structure is perfect (transmission rates T = 1) for the arbitrary angle of incidence of quasielectrons on the barrier. In the case of different values of the Fermi velocities in the barrier and out-of-barrier regions (the parameter β = υF2/υF1, which characterizes the velocity barrier, is not equal to unity), the supertunneling is observed for a certain ratio between the energy E and the barrier height U and significantly depends on β. The expression is given that determines the specified conditions for the supertunneling. In the case of equal velocities (β = 1), the supertunneling effect is observed for the quasielectron energy value E equal to half the height of the electrostatic barrier U. The analysis of the dependence of the transmission on the problem parameters is also provided.