We tackle the problem of monocular 3D reconstruction of articulated objects like humans and animals. We contribute DensePose 3D, a method that can learn such reconstructions in a weakly supervised fashion from 2D image annotations only. This is in stark contrast with previous deformable reconstruction methods that use parametric models such as SMPL pre-trained on a large dataset of 3D object scans. Because it does not require 3D scans, DensePose 3D can be used for learning a wide range of articulated categories such as different animal species. The method learns, in an end-to-end fashion, a soft partition of a given category-specific 3D template mesh into rigid parts together with a monocular reconstruction network that predicts the part motions such that they reproject correctly onto 2D DensePose-like surface annotations of the object. The decomposition of the object into parts is regularized by expressing part assignments as a combination of the smooth eigenfunctions of the Laplace-Beltrami operator. We show significant improvements compared to state-of-the-art non-rigid structure-from-motion baselines on both synthetic and real data on categories of humans and animals.