We use Gaia DR2 to measure the initial mass function (IMF) of stars within 250 pc and masses in the range 0.2 < m/Msun < 1.0, separated according to kinematics and metallicity, as determined from Gaia transverse velocity, v_T, and location on the Hertzsprung-Russell diagram (HRD). The predominant thin-disc population (v_T < 40 km/s) has an IMF similar to traditional (e.g. Kroupa 2001) stellar IMFs, with star numbers per mass interval dN/dm described by a broken power law, m^alpha, and index alpha_high=-1.99 +0.05/-0.11 above m~0.5, shallowing to alpha_low=-1.26 +0.12/-0.13 at m~<0.5. Thick-disc stars (60 km/s < v_T < 150 km/s) and stars belonging to the "high-metallicity" or "red-sequence" halo (v_T > 100 km/s or v_T > 200 km/s, and located above the isochrone on the HRD with metallicity [M.H] > -0.6) have a somewhat steeper high-mass slope, alpha_high=-2.31 +0.30/-0.81 (and a similar low-mass slope alpha_low=-1.05 +0.21/-0.65). Halo stars from the "blue sequence", which are characterised by low-metallicity ([M/H] < -0.6) , however, have a distinct, bottom-heavy IMF, well-described by a single power law with alpha=-2.17 +0.10/-0.17 over most of the mass range probed. The IMF of the low-metallicity halo is reminiscent of the Salpeter-like IMF that has been measured in massive early-type galaxies, a stellar population that, like Milky-Way halo stars, has a high ratio of alpha elements to iron, [alpha/Fe]. Blue-sequence stars are likely the debris from accretion by the Milky Way, ~10 Gyrs ago, of a moderate-mass galaxy or galaxies. These results hint at a distinct mode of star formation common to two ancient stellar populations---elliptical galaxies and galaxies accreted early-on by ours.