The conformal extension of the BMS_{3} algebra is constructed. Apart from an infinite number of "superdilatations," in order to incorporate superspecial conformal transformations, the commutator of the latter with supertranslations strictly requires the presence of nonlinear terms in the remaining generators. The algebra appears to be very rigid, in the sense that its central extensions as well as the coefficients of the nonlinear terms become determined by the central charge of the Virasoro subalgebra. The wedge algebra corresponds to the conformal group in three spacetime dimensions SO(3,2), so that the full algebra can also be interpreted as an infinite-dimensional nonlinear extension of the AdS_{4} algebra with nontrivial central charges. Moreover, since the Lorentz subalgebra [sl(2,R)] is nonprincipally embedded within the conformal (wedge) algebra, according to the conformal weight of the generators, the conformal extension of BMS_{3} can be further regarded as a W_{(2,2,2,1)} algebra. An explicit canonical realization of the conformal extension of BMS_{3} is then shown to emerge from the asymptotic structure of conformal gravity in three dimensions, endowed with a new set of boundary conditions. The supersymmetric extension is also briefly addressed.