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Mkn 421 is a nearby active galactic nucleus dominated at all wavelengths by a very broad nonthermal continuum thought to arise from a relativistic jet seen at a small angle to the line of sight. Its spectral energy distribution peaks in the X-ray and TeV γ-ray bands, where the energy output is dominated by cooling of high-energy electrons in the jet. In order to study the electron distribution and its evolution, we carried out a dedicated multi-wavelength campaign, including extensive observations by the recently launched highly sensitive hard X-ray telescope NuSTAR, between December 2012 and May 2013. Here we present some initial results based on NuSTAR data from January through March 2013, as well as calibration observations conducted in 2012. Although the observations cover some of the faintest hard X-ray flux states ever observed forMkn 421, the sensitivity is high enough to resolve intra-day spectral variability. We find that in this low state the dominant flux variations are smooth on timescales of hours, with typical intra-hour variations of ≲ 5%. We do not find evidence for either a cutoff in the hard X-ray spectrum, or a rise towards a high-energy component, but rather that at low flux the spectrum assumes a power law shape with a photon index of approximately 3. The spectrum is found to harden with increasing brightness.