  The BESS-Polar collaboration has recently performed a precise measurement of
the local antiproton flux which is consistent with a pure secondary production
of antiprotons. We constrain a possible primary component originating from dark
matter pair-annihilations. We derive limits on the annihilation cross section
which are stronger than or comparable to those from the PAMELA satellite
experiment for dark matter masses up to 200 GeV. Especially, we exclude thermal
WIMPs with masses in the range 3-20 GeV if they annihilate dominantly into
quark pairs unless their cross section is velocity suppressed.
