  We consider positive-(1,1) De Rham currents in arbitrary almost complex
manifolds and prove the uniqueness of the tangent cone at any point where the
density does not have a jump with respect to all of its values in a
neighbourhood. Without this assumption, counterexamples to the uniqueness of
tangent cones can be produced already in C^n, hence our result is optimal. The
key idea is an implementation, for currents in an almost complex setting, of
the classical blow up of curves in algebraic or symplectic geometry. Unlike the
classical approach in C^n, we cannot rely on plurisubharmonic potentials.
