In this paper, a random graph process {G(t)}_t≥1 is studied and its degree sequence is analyzed. Let {W_t}_t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with W_t edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on G(t-1), the probability that a given edge of vertex t is connected to vertex i is proportional to d_i(t-1)+δ, where d_i(t-1) is the degree of vertex i at time t-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τ_W,τ_P}, where τ_W is the power-law exponent of the initial degrees {W_t}_t≥1 and τ_P the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.