The most massive star clusters include several generations of stars with a different chemical composition (mainly revealed by an Na-O anti-correlation) while low-mass star clusters appear to be chemically homogeneous. We are investigating the chemical composition of several clusters with masses of a few 10⁴ M_⊙ to establish the lower mass limit for the multiple stellar population phenomenon. Using VLT/FLAMES spectra we determine abundances of Fe, O, Na, and several other elements (α, Fe-peak, and neutron-capture elements) in the old open cluster Berkeley 39. This is a massive open cluster: M�~�10⁴ M_⊙, approximately at the border between small globular clusters and large open clusters. Our sample size of about 30 stars is one of the largest studied for abundances in any open cluster to date, and will be useful to determine improved cluster parameters, such as age, distance, and reddening when coupled with precise, well-calibrated photometry. We find that Berkeley 39 is slightly metal-poor, ⟨[Fe/H]⟩�=�−0.20, in agreement with previous studies of this cluster. More importantly, we do not detect any star-to-star variation in the abundances of Fe, O, and Na within quite stringent upper limits. The rms scatter is 0.04, 0.10, and 0.05 dex for Fe, O, and Na, respectively. This small spread can be entirely explained by the noise in the spectra and by uncertainties in the atmospheric parameters. We conclude that Berkeley 39 is a single-population cluster.