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Let M be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1, n ≤ 7, and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H, such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and 
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