During hard X-ray observations of the Norma spiral arm region by the Nuclear Spectroscopic Telescope Array (NuSTAR) in 2013 February, a new transient source, NuSTAR J163433-4738.7, was detected at a significance level of 8σ in the 3-10 keV bandpass. The source is consistent with having a constant NuSTAR count rate over a period of 40 ks and is also detected simultaneously by Swift at lower significance. The source is not significantly detected by NuSTAR, Swift, or Chandra in the days before or weeks after the discovery of the transient, indicating that the strong X-ray activity lasted between ~0.5 and 1.5 days. Near-infrared imaging observations were carried out before and after the X-ray activity, but we are not able to identify the counterpart. The combined NuSTAR and Swift energy spectrum is consistent with a power law with a photon index of (90% confidence errors), a blackbody with kT = 1.2 ± 0.3 keV, or a Bremsstrahlung model with keV. The reduced-χ2 values for the three models are not significantly different, ranging from 1.23 to 1.44 for 8 degrees of freedom. The spectrum is strongly absorbed with cm-2, cm-2, and cm-2, for the power-law, blackbody, and Bremsstrahlung models, respectively. Although the high column density could be due to material local to the source, it is consistent with absorption from interstellar material along the line of sight at a distance of 11 kpc, which would indicate an X-ray luminosity >1034 erg s-1. Although we do not reach a definitive determination of the nature of NuSTAR J163433-4738.7, we suggest that it may be an unusually bright active binary or a magnetar.