The Sun has been observed by Fermi-LAT as a surprisingly bright source of gamma rays up to hundreds of GeV. The gamma rays can be attributed to interactions of cosmic rays in the solar atmosphere. Certain models of solar dark matter capture and annihilation also predict steady gamma-ray emission from the Sun. Interestingly, at GeV energies the flux observed by Fermi-LAT is much brighter ($>$ factor of 6) and spectrally harder than predicted from cosmic-ray interactions. The flux also shows a dependence on solar activity, with the highest energy photons occurring during the solar minimum and exhibiting no spectral cutoff up to at least 200 GeV. Observations of the Sun above 1 TeV may shed light on the mechanisms which produce the unexplained gamma rays. Non-observations of gamma-rays $>$1 TeV can be used to constrain cosmic-ray interactions, and provide limits on dark matter annihilation inside the Sun. The High Altitude Water Cherenkov Observatory (HAWC) is currently the only detector capable of monitoring the Sun at energies above 1 TeV. HAWC observations placed strong constraints on steady TeV emission from the Sun during the active phase of solar cycle 24, demonstrating sensitivity to an energy flux of $\sim10^{-12}$ TeV cm $^{-2}$ s$^{-1}$. The Sun entered a new solar minimum beginning 2018. We present new results from 1 year of data collected by HAWC during the ongoing solar minimum and discuss the implications for models of gamma-ray emission from the Sun.