We report on measurements of cross-species thermalization inside a magnetically trapped spin-polarized mixture of $^{87}$Rb and $^7$Li atoms with both atoms in their respective low field seeking magnetic substates $|F=2,m_F=2right>$. Measurement of the thermalization velocity in the ultracold regime below $10 mu$K allows for the derivation of the absolute value of the pure triplet s-wave scattering length governing the interaction. We find $|a_{7,87}|=(59pm19) a_{rm B}$. We propose to study both species in the condensed regime to derive the sign of $a_{7,87}$. In this context, we present numerical solutions to the coupled Gross-Pitaevskii equation based on the hypothesis of a positive sign. According to the simulations, phase separation of the Li and Rb $|2,2right>$ clouds occurs along with a mean-field stabilization allowing for larger atom numbers of condensed $^7$Li atoms before collapse sets in. Observation of this mean-field stabilization would directly fix the sign of $a_{7,87}$. We discuss our results in the light of this proposal.
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