The space of currents (forms with distribution coefficients) on a complex manifold contains a distinguished cone of positive currents which have been defined and studied by Lelong. A subcone of this cone is generated locally by those currents which correspond to integration over complex subvarieties. Lelong has conjectured that these are precisely the positive currents which are d-closed and whose Lelong numbers, or densities, are locally bounded away from zero [6]. We prove this conjecture in Structure Theorem 6 below. For currents of degree (1, 1) (the hypersurface case) Lelong's conjecture follows immediately from a structure theorem of Bombieri [1] and [2] and Lemma 7 below. One of the authors has verified the conjecture for any degree under the assumption that the densities take integer values [4]. In this paper we show that Bombieri's Structure Theorem for currents of degree (i, 1) can be used to prove the complete conjecture of Lelong. Let ~,.k(~2) denote the space of currents of degree (k, k) on (~ open c ~". That is, u e ~k,k (O) if U = ~, U H d z I/x d5 s where each u H e ~ ' (O) Ill=IJi=k is a distribution on ~. I denotes the multi-index (i~,..., ik), and dz~= dzil A ... A d z i . If ~ belongs to the space ~2,_2k(Q) of compactly supported C ~ (2n-2k)forms on ~, then u(~) is by definition ~ A u(1). The usual differential operators d, r~, and • defined on ~k.k(O) extend to