The measurement of the structure of stellar populations in the Milky Way disc places fundamental constraints on models of galaxy formation and evolution. Previously, the disc's structure has been studied in terms of populations defined geometrically and/or chemically, but a decomposition based on stellar ages provides a more direct connection to the history of the disc, and stronger constraint on theory. Here, we use positions, abundances and ages for 31 244 red giant branch stars from the Sloan Digital Sky Survey (SDSS)-APOGEE survey, spanning 3 < Rgc < 15 kpc, to dissect the disc into mono-age and mono-[Fe/H] populations at low and high [α/Fe]. For each population, with Δage < 2 Gyr and Δ[Fe/H] < 0.1 dex, we measure the structure and surface-mass density contribution. We find that low [α/Fe] mono-age populations are fit well by a broken exponential, which increases to a peak radius and decreases thereafter. We show that this profile becomes broader with age, interpreted here as a new signal of disc heating and radial migration. High [α/Fe] populations are well fit as single exponentials within the radial range considered, with an average scalelength of 1.9 ± 0.1 kpc. We find that the relative contribution of high to low [α/Fe] populations at R0 is f_Σ = 18 per cent ± 5 per cent; high [α/Fe] contributes most of the mass at old ages, and low [α/Fe] at young ages. The low and high [α/Fe] populations overlap in age at intermediate [Fe/H], although both contribute mass at R0 across the full range of [Fe/H]. The mass-weighted scaleheight hZ distribution is a smoothly declining exponential function. High [α/Fe] populations are thicker than low [α/Fe], and the average hZ increases steadily with age, between 200 and 600 pc.