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This thesis presents a measurement of the charm mixing parameters $x_D$ and $y_D$ in prompt $D^{0} \rightarrow K_S^0 \pi^+ \pi^-$ decays using 1 fb$^{-1}$ of data collected by the LHCb experiment during 2011. \par Mixing in charm is predicted to be small within the Standard Model, but there are significant uncertainties associated with calculating the long range contributions to the decay. Recent measurements made by LHCb and others have confirmed that mixing in charm exists at a rate of less than 1 %. With LHCb due to collect more data and Belle II being commissioned, the reduction of systematic uncertainties will become increasingly important. The $D^{0} \rightarrow K_S^0 \pi^+ \pi^-$ decay provides sensitivity to both the magnitude and relative sign between the mixing parameters. It is also one of the few channels that can measure $x_D$ directly. It is therefore crucial to study this mode in detail as more data becomes available. The work presented in this thesis utilises a model-independent description of the $K_S^0 \pi^+ \pi^-$ Dalitz plot decay for the first time in the context of charm mixing. Previous mixing measurements with this final state have used a Dalitz plot amplitude model, and the associated systematic uncertainty is not straight forward to estimate or control. In its place, this analysis uses external, statistically-limited measurements of the strong-phase difference between $D^{0}$ and $\bar{D}^0$ obtained by CLEO as input. In addition, a data-driven technique is used to correct for decay time biases induced by the selection removing any systematic effects due to extracting this from simulated data. As the amount of available data increases, both of these techniques will become vital to improving our understanding of mixing in charm. \par In the CP convention used by BaBar and adopted for this thesis, the measured mixing parameters are \begin{equation*} \begin{split} x_D = & \, -(0.863 \pm 0.527\, (\text{stat.}) \pm 0.171\, (\text{syst.}))\, \%, \\ y_D = & \, -(0.026 \pm 0.463\, (\text{stat.}) \pm 0.134\, (\text{syst.}))\, \%. \\ \end{split} \end{equation*} Both $x_D$ and $y_D$ are consistent with the current world averages.