We use Gaia photometry and astrometry to estimate the vertical spatial structure of the Milky Way at the Solar radius, formally accounting for sample incompleteness (the selection function) and parallax measurement uncertainty. Our results show impressive precision demonstrating the power of the Gaia data. However, systematic errors dominate the parameter value uncertainties. We thoroughly test and quantify the impacts of all systematic uncertainties. The vertical tracer density is modelled as a sum of two exponential profiles for the thin and thick discs, together with a spherically symmetric power law for the stellar halo. We constrain the thin disc scale height as ${h_\mathrm{Tn}=260 \pm 3\, (\mathrm{stat}) \pm 26\, \mathrm{pc}\, (\mathrm{sys})}$ and thick disc ${h_\mathrm{Tk}=693 \pm 7 \, (\mathrm{stat}) \pm 121\, \mathrm{pc}\, (\mathrm{sys})}$. For the halo, we obtain a power-law profile with $n_\mathrm{H}=3.543\pm 0.023 \, (\mathrm{stat}) \pm 0.259\, (\mathrm{sys})$. We infer a local stellar mass density for non-compact object stars of ${\rho _\mathrm{local}^{*} = 3.66\pm 0.03\, (\mathrm{stat})\pm 0.52 \times 10^{-2}\, \mathrm{M}_\odot \, \mathrm{pc}^{-3}\, (\mathrm{sys})}$ and surface density of ${\Sigma _\mathrm{local}^{*} = 23.17\pm 0.08\, (\mathrm{stat})\pm 2.43\, \mathrm{M}_\odot \, \mathrm{pc}^{-2}\, (\mathrm{sys})}$. We find asymmetries above and below the disc with longer disc scale heights in the north but a flatter halo in the south at the ≲ 10 per cent level.