We describe a procedure that allows continuously tuning the average degree $\langle k \rangle$ of uncorrelated networks with power-law degree distribution $p(k)$. Inn order to do this, we modify the low-$k$ region of $p(k)$, while preserving the large-$k$ tail up to a cutoff. Then, we use the modified $p(k)$ to obtain the degree sequence required to construct networks through the configuration model. We analyze the resulting nearest-neighbor degree and local clustering to verify the absence of $k$-dependencies. Finally, a further modification is introduced to eliminate the sample fluctuations in the average degree.