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 axionlike particle dark matter is relatively unexplored, particularly at masses m_{a}≲1 μeV. ABRACADABRA is a new experimental program to search for axion dark matter over a broad range of masses, 10^{-12}≲m_{a}≲10^{-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 axionlike cosmic dark matter and set 95% C.L. upper limits on the axion-photon coupling between g_{aγγ}<1.4×10^{-10} and g_{aγγ}<3.3×10^{-9} GeV^{-1} over the mass range 3.1×10^{-10}-8.3×10^{-9} eV. These results are competitive with the most stringent astrophysical constraints in this mass range.