In section II.5 Hartshorne describes the notion of a sheaf $\tilde{M}$ associated to a module M over a ring A.
Now in the case $A:= \mathbb{C}$, we have $\operatorname{Spec}(\mathbb{C}) = {0}$. Unless I'm mistaken (very probable) since there is only one stalk, which correponds to the entire sheaf, then $$\tilde{M} \cong \{ s: 0 \rightarrow \mathbb{C} | (\exists m \in M) s=m \} \cong \underline{M}$$, the constant sheaf?
Have I made a mistake, or am I right? :)
You should write $\mathrm{Spec}(\mathbb{C}) = \{0\}$, the set with one element being the zero ideal. To answer your question, yes, as in fact every sheaf on a one-point space is necessarily the constant sheaf. This is simply because, the space having one point, there is literally only one group of sections, and that is the group with which the constant sheaf is associated, tautologically.