I don't yet have a reference, but it seems that the result was first proved by Fichtenholz and Sierpiński, independently from each other. This should be mentioned in a remark to Problem 12 in:
R. Sikorski, Real Functions, Vol. 1, PWN: Warsaw, 1958 (in Polish),
at least according to the historical remark on p. 28 in:
P. Lorenc and R. Wituła, Darboux property of the nonatomic $\sigma$-additive positive and finite dimensional vector measures, Matematyka Stosowana 3 (2013), 25-36.
Unfortunately, I couldn't retrieve either a hard copy of Sikorski's book or simply a scan of the relevant page(s).
Update (Nov 27, 2015). Thanks to Martin Sleziak and Jacek Jendrej, I found out that the "remark to Problem 12" referred to by Lorenc and Wituła is actually a footnote on p. 225 of Sikorski's book.
Now, Problem 12 reads (caveat: the translation is mine and may not be very precise, as my Polish is null), "Let $\mu$ be a non-atomic measure and let $0 < s < \mu(A) < \infty$. 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 does make reference to "Sierpiński [7] and Fichtenholz [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?