Measurements are presented of $K~0$ meson and $\Lambda$ baryon production in deep-inelastic positron-proton scattering (DIS) in the kinematic range $10 < Q~2 < 70\,$GeV$~2$ and $10~{-4} < x < 10~{-2}$. The measurements, obtained using the H1 detector at the HERA collider, are discussed in the light of possible mechanisms for increased strangeness production at low Bjorken-$x$. Comparisons of the $x_F$ spectra, where $x_F$ is the fractional longitudinal momentum in the hadronic centre-of-mass frame, with results from electron-positron annihilation are made. The $x_F$ spectra and the $K~0$ ``seagull'' plot are compared with previous DIS results. The mean $K~0$ and $\Lambda$ multiplicities are studied as a function of the centre-of-mass energy $W$ and are observed to be consistent with a logarithmic increase with $W$ when compared with previous measurements. A comparison of the levels of strangeness production in diffractive and non-diffractive DIS is made. An upper limit of $0.9\,$nb, at the $95\%$ confidence level, is placed on the cross-section for QCD instanton induced events.