A new superconductor with an original structure, Sr4-xBaTlCu2CO3O7 has been isolated for 1 less-than-or-equal-to x less-than-or-equal-to 2. It crystallizes in a A-type space group with a=3.84 angstrom almost-equal-to a(p), b almost-equal-to 8 x a(p), c almost-equal-to 17.0 angstrom. The HREM study of this new cuprate shows that it derives from the Tl0.5Pb0.5Sr4Cu2CO3O7 structure by a shearing mechanism. Indeed, it can be described as ribbons of the latter structure, four octahedra thick, shifted c/2 with respect to each other. It results in infinite single perovskite layers waving along the (001) plane and connected through flat mixed layers [(TlO)4(CO)4]infinity. This oxycarbonate can also be described as a 1201-type structure in which the [TlO]infinity layers are replaced by mixed [(TlO)4(CO)4]infinity layers. The susceptibility measurements show that this cuprate exhibits a significantly higher critical temperature than the 1201-cuprates. The T(c onset) of the synthesized phase is 56 K and it can be increased up to 62 K by hydrogen annealing; in the same way, one observes a large Meissner volume fraction of 35% at 4.2 K.