  We study the accuracy with which the WIMP mass could be determined by a
superCDMS-like direct detection experiment, given optimistic assumptions about
the detector set-up and WIMP properties. We consider WIMPs with an interaction
cross-section of \sigma_{\rm p} = 10^{-7} {\rm pb} (just below current
exclusion limits) and assume, initially, that the local WIMP velocity
distribution and density are known and that the experiment has negligible
background. For light WIMPs (mass significantly less than that of the target
nuclei) small variations in the WIMP mass lead to significant changes in the
energy spectrum. Conversely for heavy WIMPs the energy spectrum depends only
weakly on the WIMP mass. Consequently it will be far easier to measure the WIMP
mass if it is light than if it is heavy. With exposures of {\cal E}= 3 \times
10^{3}, 3 \times 10^{4} and 3 \times 10^{5} {\rm kg day} (corresponding,
roughly, to the three proposed phases of SuperCDMS) it will be possible, given
the optimistic assumptions mentioned above, to measure the mass of a light WIMP
with an accuracy of roughly 25%, 15% and 2.5 % respectively. These numbers
increase with increasing WIMP mass, and for heavy WIMPs, m_{\chi} > {\cal
O}(500 {\rm GeV}), even with a large exposure it will only be possible to place
a lower limit on the mass. Finally we discuss the validity of the various
assumptions made, and the consequences if these assumptions are not valid. In
particular if the local WIMP distribution is composed of a number of discrete
streams it will not be possible to determine the WIMP mass.
