We present numerical evidence from Monte-Carlo simulations that the superfluid-insulator quan- tum phase transition of interacting bosons subject to strong disorder in one dimension is controlled by a novel critical point. At the critical point the distribution of superfluid stiffness over disorder realizations develops a power-law tail reflecting a universal distribution of weak links. The Luttinger parameter on the other hand does not take on a universal value at this critical point, in marked contrast to the known Berezinskii-Kosterlitz-Thouless-like superfluid-insulator transition in weakly disordered systems. We develop a finite-size scaling procedure which allows us to directly compare the numerical results from systems of linear size up to 1024 sites with theoretical predictions ob- tained in [PRL 93, 150402 (2004)], using a strong disorder renormalization group approach. The data shows good agreement with the scaling expected at the strong-disorder critical point.