The enhanced electric field at plasmonic resonances in nanoscale antennas can lead to efficient harmonic generation, especially when the plasmonic geometry is asymmetric on either inter-particle or intra-particle levels. The planar Archimedean nanospiral offers a unique geometrical asymmetry for second-harmonic generation (SHG) because the SHG results neither from arranging centrosymmetric nanoparticles in asymmetric groupings, nor from non-centrosymmetric nanoparticles that retain a local axis of symmetry. Here we report forward SHG from planar arrays of Archimedean nanospirals using 15 fs pulse from a Ti:sapphire oscillator tuned to 800 nm wavelength. The measured harmonic-generation efficiencies are 2.6*10-9, 8*10-9 and 1.3*10-8 for left-handed circular, linear, and right-handed circular polarizations, respectively. The uncoated nanospirals are stable under average power loading of as much as 300 uW per nanoparticle. The nanospirals also exhibit a selective conversion between polarization states. These experiments show that the intrinsic asymmetry of the nanospirals results in a highly efficient, two-dimensional harmonic generator that can be incorporated into metasurface optics.