Let Mⁿ (n ≥ 3) be an n-dimensional complete hypersurface in a real space form N(c) (c ≥ 0). We prove that if the sectional curvature $K_M$ of M satisfies the following pinching condition: $c+\delta <K_M\le c+1$, where δ = 1/5 for n ≥ 4 and δ = 1/4 for n = 3, then there are no stable currents (or stable varifolds) in M. This is a positive answer to the well-known conjecture of Lawson and Simons.