I don't yet have a reference, but it seems that the result wasmight have been first proved by Fichtenholz and Sierpiński, independently from each other. This should be mentioned in a remark to Problem 12 in:
Now, Problem 12 reads, "If $\mu$ is a non-atomic measure and $0 < s < \mu(A) < \infty$, then there exists $B \subseteq A$ such that $\mu(B) = s$", so we are really talking of Theorem 1 in the OP. And the footnote on p. 225 really makes reference to "Sierpiński [7] and Fichtenholz 1"[${}$1]". But the copy of Sikorski's book I could retrieve is incomplete and doesn't include the bibliography... Could anyone having access to a complete copy fill in this answer?
Update (Dec 0708, 2015). Here is some more evidence that the common (?) attribution of Theorem 1 in the OP to W. Sierpiński may be wrong.