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In this paper we prove that,under condition,a strong law of large numbers holds for a class of super-diffusions Xcorresponding tothe eolution equation θtυt=Lυt+βυt-ψ(υt)on a bounded domain D in Rd,where the branching mechanism ψ(χ,λ)=αλ2+∫0∞(e-λΥ)(x,dΥ)satisfies sup x∈D∫0∞(Υ∧Υ2)η(x,dΥ)<∞.