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We investigate the detection efficiency of a spiral layout of a superconducting nanowire single-photon detector. The design is less susceptible to the critical current reduction in sharp turns of the nanowire than the conventional meander design. Detector samples with different nanowire widths from 100 to 300 nm are patterned from a 4-nm-thick NbN film deposited on sapphire substrates. The critical current IC at 4.2 K for spiral, meander, and single-bridge structures is measured and compared. On the 100-nm-wide samples, the detection efficiency is measured in the wavelength range 400-1700 nm and the spectral bandwidth of the intrinsic detection efficiency is determined. In the optical range, the spiral detector reaches a detection efficiency of 27.6%, which is ~ 1.5 times the value of the meander. In the infrared range the detection efficiency of the spiral superconducting nanowire single-photon detector is more than doubled.