let $f$ be a totally real number field and let $p$ be a finite prime of $f$, such that $p$ splits completely in the finite abelian extension $h$ of $f$. stark has proposed a conjecture stating the existence of a $p-unit$ in $f$ with absolute values at the places above $p$ specified in terms of the values at zero of the partial zeta functions associated to $h/f$. gross proposed a refinement of stark's conjecture which gives a conjectural formula for the image of stark's unit in $f_p^\times/ \widehat e$, where $f_p$ denotes the completion of $f$ at $p$ and $\widehat e$ denotes the topological closure of the group of totally positive units of $f$. we propose a further refinement of gross' conjecture by proposing a conjectural formula for the exact value of stark's unit in $f_p^\times$.