NuSTAR observed G1.9+0.3, the youngest known supernova remnant in the Milky Way, for 350 ks and detected emission up to $sim$30 keV. The remnants X-ray morphology does not change significantly across the energy range from 3 to 20 keV. A combined fit between NuSTAR and CHANDRA shows that the spectrum steepens with energy. The spectral shape can be well fitted with synchrotron emission from a power-law electron energy distribution with an exponential cutoff with no additional features. It can also be described by a purely phenomenological model such as a broken power-law or a power-law with an exponential cutoff, though these descriptions lack physical motivation. Using a fixed radio flux at 1 GHz of 1.17 Jy for the synchrotron model, we get a column density of N$_{rm H}$ = $(7.23pm0.07) times 10^{22}$ cm$^{-2}$, a spectral index of $alpha=0.633pm0.003$, and a roll-off frequency of $ u_{rm rolloff}=(3.07pm0.18) times 10^{17}$ Hz. This can be explained by particle acceleration, to a maximum energy set by the finite remnant age, in a magnetic field of about 10 $mu$G, for which our roll-off implies a maximum energy of about 100 TeV for both electrons and ions. Much higher magnetic-field strengths would produce an electron spectrum that was cut off by radiative losses, giving a much higher roll-off frequency that is independent of magnetic-field strength. In this case, ions could be accelerated to much higher energies. A search for $^{44}$Ti emission in the 67.9 keV line results in an upper limit of $1.5 times 10^{-5}$ $,mathrm{ph},mathrm{cm}^{-2},mathrm{s}^{-1}$ assuming a line width of 4.0 keV (1 sigma).
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