We present HST/STIS time-series spectroscopy of the central star of the "Cat's Eye" planetary nebula NGC 6543. Intensive monitoring of the UV lines over a 5.8 hour period reveals well defined details of large-scale structure in the fast wind, which are exploited to provide new constraints on the rotation rate of the central star. We derive characteristics of the line profile variability that support a physical origin due to co-rotating interaction regions (CIRs) that are rooted at the stellar surface. The recurrence time of the observed spectral signatures of the CIRs is used to estimate the rotation period of the central star and, adopting a radius between 0.3 and 0.6 \Rsun constrains the rotational velocity to the range 54 \leq v_{rot} \leq 108 \kms. The implications of these results for single star evolution are discussed based on models calculated here for low-mass stars. Our models predict a sub-surface convective layer in NGC 6543 which we argue to be causally connected to the occurrence of structure in the fast wind.