We study the mixing of the Gluino-Glue operator in ${cal N}$=1 Supersymmetric Yang-Mills theory (SYM), both in dimensional regularization and on the lattice. We calculate its renormalization, which is not only multiplicative, due to the fact that this operator can mix with non-gauge invariant operators of equal or, on the lattice, lower dimension. These operators carry the same quantum numbers under Lorentz transformations and global gauge transformations, and they have the same ghost number. We compute the one-loop quantum correction for the relevant two-point and three-point Greens functions of the Gluino-Glue operator. This allows us to determine renormalization factors of the operator in the $overline{textrm{MS}}$ scheme, as well as the mixing coefficients for the other operators. To this end our computations are performed using dimensional and lattice regularizations. We employ a standard discretization where gluinos are defined on lattice sites and gluons reside on the links of the lattice; the discretization is based on Wilsons formulation of non-supersymmetric gauge theories with clover improvement. The number of colors, $N_c$, the gauge parameter, $beta$, and the clover coefficient, $c_{rm SW}$, are left as free parameters.
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