The apparent end of the internally generated Martian magnetic field at 3.6-4.1 Ga has been linked to insufficient core cooling. While the dynamo cessation time is a key event in the Martian history, a large range of solutions exist to satisfy this observation, limiting the insight it provides on the Martian interior. We produce a suite of models improving our understanding of the evolution of Mars in three keys areas. Firstly, we account for thermal stratification in the core, producing improved estimates on the core thermal structure. Secondly, we match estimates for the present-day areotherm. Finally, we consider recent experimental data for the core thermal conductivity, k_c, much lower than previously thought for Mars. In order to yield a consistent dynamo cessation time we require k_c<18 Wm^-1K^-1. Furthermore, the majority of our models indicate the core is fully conductive at present and too hot to form any solid.