This Letter reports on the first results from the Taiwan Axion Search Experiment with a Haloscope, a search for axions using a microwave cavity at frequencies between 4.707 50 and 4.798 15 GHz. Apart from the nonaxion signals, no candidates with a significance of more than 3.355 were found. The experiment excludes models with the axion-two-photon coupling |g_{aγγ}|≳8.1×10^{-14} GeV^{-1}, a factor of eleven above the benchmark Kim-Shifman-Vainshtein-Zakharov model, in the mass range 19.4687<m_{a}<19.8436 μeV. It is also the first time that a haloscope experiment places constraints on g_{aγγ} in the mass region of 19.4687<m_{a}<19.7639 μeV, reaching a sensitivity 3 orders of magnitude better than the limits obtained by nonhaloscope experiments.