In the cascading gravity brane-world scenario, our 3-brane lies within a succession of lower codimension branes, each with their own induced gravity term, embedded into each other in a higher dimensional space-time. In the (6 + 1)-dimensional version of this scenario, we show that a 3-brane with tension remains flat, at least for sufficiently small tension that the weak-field approximation is valid. The bulk solution is singular nowhere and remains in the perturbative regime everywhere.