We report the discovery of two transiting Neptunes by the HATSouth survey. The planet HATS-37Ab has a mass of 0.099 +- 0.042 M_J (31.5 +- 13.4 M_earth) and a radius of 0.606 +- 0.016 R_J, and is on a P = 4.3315 days orbit around a V = 12.266 mag, 0.843 M_sun star with a radius of 0.877 R_sun. We also present evidence that the star HATS-37A has an unresolved stellar companion HATS-37B, with a photometrically estimated mass of 0.654 M_sun.The planet HATS-38b has a mass of 0.074 +- 0.011 M_J (23.5 +- 3.5 M_earth) and a radius of 0.614 +- 0.017 R_J, and is on a P = 4.3750 days orbit around a V = 12.411 mag, 0.890 M_sun star with a radius of 1.105 R_sun. Both systems appear to be old, with isochrone-based ages of 11.46 +0.79-1.45 Gyr, and 11.89 +- 0.60 Gyr, respectively. Both HATS-37Ab and HATS-38b lie in the Neptune desert and are thus examples of a population with a low occurrence rate. They are also among the lowest mass planets found from ground-based wide-field surveys to date.
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