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This article proposes a new algorithm to compute the projection on the set of images whose total variation is bounded by a constant. The projection is computed through a dual formulation that is solved by first order non-smooth optimization methods. This yields an iterative algorithm that applies iterative soft thresholding to the dual vector field, and for which we establish convergence rate on the primal iterates. This projection algorithm can then be used as a building block in a variety of applications such as solving inverse problems under a total variation constraint, or for texture synthesis. Numerical results are reported to illustrate the usefulness and potential applicability of our TV projection algorithm on various examples including denoising, texture synthesis, inpainting, deconvolution and tomography problems. We also show that our projection algorithm competes favorably with state-of-the-art TV projection methods in terms of convergence speed.