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We extend the BMS(4) group by adding logarithmic supertranslations. This is done by relaxing the boundary conditions on the metric and its conjugate momentum at spatial infinity in order to allow logarithmic terms of carefully designed form in the asymptotic expansion, while still preserving finiteness of the action. Standard theorems of the Hamiltonian formalism are used to derive the (finite) generators of the logarithmic supertranslations. As the ordinary supertranslations, these depend on a function of the angles. Ordinary and logarithmic supertranslations are then shown to form an abelian subalgebra with non-vanishing central extension. Because of this central term, one can make nonlinear redefinitions of the generators of the algebra so that the pure supertranslations ($\ell >1$ in a spherical harmonic expansion) and the logarithmic supertranslations have vanishing brackets with all the Poincaré generators, and, in particular, transform in the trivial representation of the Lorentz group. The symmetry algebra is then the direct sum of the Poincaré algebra and the infinite-dimensional abelian algebra formed by the pure supertranslations and the logarithmic supertranslations (with central extension). The pure supertranslations are thus completely decoupled from the standard Poincaré algebra in the asymptotic symmetry algebra. This implies in particular that one can provide a definition of the angular momentum which is manifestly free from supertranslation ambiguities. An intermediate redefinition providing a partial decoupling of the pure and logarithmic supertranslations is also given.