  We obtain explicit expressions for one unknown thermal coefficient (among the
conductivity, mass density, specific heat and latent heat of fusion) of a
semi-infinite material through the one-phase fractional Lam\'e-Clapeyron-Stefan
problem with an over-specified boundary condition on the fixed face $x=0$. The
partial differential equation and one of the conditions on the free boundary
include a time Caputo's fractional derivative of order $\alpha \in (0,1) $.
Moreover, we obtain the necessary and sufficient conditions on data in order to
have a unique solution by using recent results obtained for the fractional
diffusion equation exploiting the properties of the Wright and Mainardi
functions, given in Roscani - Santillan Marcus, Fract. Calc. Appl. Anal., 16
(2013), 802-815, Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237-249, and
Voller, Int. J. Heat Mass Transfer, 74 (2014), 269-277. This work generalizes
the method developed for the determination of unknown thermal coefficients for
the classical Lam\'e-Clapeyron-Stefan problem given in Tarzia, Adv. Appl.
Math., 3 (1982), 74-82, which are recovered by taking the limit when the order
$\alpha\nearrow 1$.
