The axion is a promising dark matter candidate, which was originally proposed to solve the strong-CP problem in particle physics. To date, the available parameter space for axion and axion-like particle dark matter is relatively unexplored, particularly at masses $m_alesssim1,mu$eV. ABRACADABRA is a new experimental program to search for axion dark matter over a broad range of masses, $10^{-12}lesssim m_alesssim10^{-6}$ eV. ABRACADABRA-10 cm is a small-scale prototype for a future detector that could be sensitive to the QCD axion. In this Letter, we present the first results from a 1 month search for axions with ABRACADABRA-10 cm. We find no evidence for axion-like cosmic dark matter and set 95% C.L. upper limits on the axion-photon coupling between $g_{agammagamma}<1.4times10^{-10}$ GeV$^{-1}$ and $g_{agammagamma}<3.3times10^{-9}$ GeV$^{-1}$ over the mass range $3.1times10^{-10}$ eV - $8.3times10^{-9}$ eV. These results are competitive with the most stringent astrophysical constraints in this mass range.
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