  We associate a C*-algebra $\widetilde{\mathcal{O}}_{\textsf{X}}$ with a
subshift over an arbitrary, possibly infinite, alphabet. We show that
$\widetilde{\mathcal{O}}_{\textsf{X}}$ is a full invariant for topological
conjugacy of the subshifts of Ott, Tomforde, and Willis. When the alphabet is
countable, we show that $\widetilde{\mathcal{O}}_{\textsf{X}}$ is an invariant
for isometric conjugacy of subshifts with the product metric. For a suitable
partial action associated with a subshift over a countable alphabet, we show
that $\widetilde{\mathcal{O}}_{\textsf{X}}$ is also an invariant for continuous
orbit equivalence. Additionally, we give a concrete way to compute the K-theory
of $\widetilde{\mathcal{O}}_{\textsf{X}}$ and illustrate it with two examples.
