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The tidal disruption of the Sagittarius dwarf galaxy has generated a spectacular stream of stars wrapping around the entire Galaxy. We use data from $Gaia$ and the H3 Stellar Spectroscopic Survey to identify 823 high-quality Sagittarius members based on their angular momenta. The H3 Survey is largely unbiased in metallicity, and so our sample of Sagittarius members is similarly unbiased. Stream stars span a wide range in [Fe/H] from $-0.2$ to $\approx -3.0$, with a mean overall metallicity of $\langle$[Fe/H]$\rangle=-0.99$. We identify a strong metallicity-dependence to the kinematics of the stream members. At [Fe/H]$\gt -0.8$ nearly all members belong to the well-known cold ($σ_v \lt 20$ km/s) leading and trailing arms. At intermediate metallicities ($-1.9 \lt$[Fe/H]$\lt -0.8$) a significant population (24$\%$) emerges of stars that are kinematically offset from the cold arms. These stars also appear to have hotter kinematics. At the lowest metallicities ([Fe/H]$\lesssim-2$), the majority of stars (69$\%$) belong to this kinematically-offset diffuse population. Comparison to simulations suggests that the diffuse component was stripped from the Sagittarius progenitor at earlier epochs, and therefore resided at larger radius on average, compared to the colder metal-rich component. We speculate that this kinematically diffuse, low metallicity, population is the stellar halo of the Sagittarius progenitor system.