We present an analysis of the Hubble diagram for 12 normal Type Ia supernovae (SNe Ia) observed in the near-infrared (NIR) J and H bands. We select SNe exclusively from the redshift range 0.03 < z < 0.09 to reduce uncertainties coming from peculiar velocities while remaining in a cosmologically well-understood region. All of the SNe in our sample exhibit no spectral or B-band light-curve peculiarities and lie in the B-band stretch range of 0.8–1.15. Our results suggest that SNe Ia observed in the NIR are the best known standard candles. We fit previously determined NIR light-curve templates to new high-precision data to derive peak magnitudes and to determine the scatter about the Hubble line. Photometry of the 12 SNe is presented in the natural system. Using a standard cosmology of (H₀, Ω_m, Ω_Λ) = (70, 0.27, 0.73), we find a median J-band absolute magnitude of M_J = −18.39 with a scatter of σ_J = 0.116 and a median H-band absolute magnitude of M_H = −18.36 with a scatter of σ_H = 0.085. The scatter in the H band is the smallest yet measured. We search for correlations between residuals in the J- and H-band Hubble diagrams and SN properties, such as SN colour, B-band stretch and the projected distance from the centre of the host galaxy. The only significant correlation is between the J-band Hubble residual and the J − H pseudo-colour. We also examine how the scatter changes when fewer points in the NIR are used to constrain the light curve. With a single point in the H band taken anywhere from 10 d before to 15 d after B-band maximum light and a prior on the date of H-band maximum set from the date of B-band maximum, we find that we can measure distances to an accuracy of 6 per cent. The precision of SNe Ia in the NIR provides new opportunities for precision measurements of both the expansion history of the universe and peculiar velocities of nearby galaxies.