  We present experimental observations of seven vibrational levels $v'=20-26$
of the $1^{3}\Sigma_{g}^{+}$ excited state of Li$_2$ molecules by the
photoassociation (PA) of a degenerate Fermi gas of $^6$Li atoms. For each
vibrational level, we resolve the rotational structure using a Feshbach
resonance to enhance the PA rates from p-wave collisions. We also, for the
first time, determine the spin-spin and spin-rotation interaction constants for
this state. The absolute uncertainty of our measurements is $\pm 0.00002$
cm$^{-1}$ ($\pm 600$ kHz). We use these new data to further refine an analytic
potential for this state.
