  We present the first study on the amplification of magnetic fields by the
turbulent dynamo in the highly subsonic regime, with Mach numbers ranging from
$10^{-3}$ to $0.4$. We find that for the lower Mach numbers the saturation
efficiency of the dynamo, $(E_{\mathrm{mag}}/E_{\mathrm{kin}})_{\mathrm{sat}}$,
increases as the Mach number decreases. Even in the case when injection of
energy is purely through longitudinal forcing modes,
$(E_{\mathrm{mag}}/E_{\mathrm{kin}})_{\mathrm{sat}}$ $\gtrsim 10^{-2}$ at a
Mach number of $10^{-3}$. We apply our results to magnetic field amplification
in the early Universe and predict that a turbulent dynamo can amplify
primordial magnetic fields to $\gtrsim$ $10^{-16}$ Gauss on scales up to 0.1 pc
and $\gtrsim$ $10^{-13}$ Gauss on scales up to 100 pc. This produces fields
compatible with lower limits of the intergalactic magnetic field inferred from
blazar $\gamma$-ray observations.
