The last few lines provide us with all the evidence that we need:
Whene’er I passed her; but who passed without
Much the same smile? This grew; I gave commands;
Then all smiles stopped together. There she stands
As if alive . . .
The two telling—and chilling —phrases are "Then all smiles stopped together" and "As if alive" (my emphasis). The speaker would only say "as if alive" if she weren't actually alive. That's the short answer. However, it has required me to begin at the end, and a little context might shed some more light on her fate.
's "The Last Duchess" is an example of a dramatic monologue
; the narrator is presumably the husband of the titular Duchess. This means that he is a Duke. We know, too, that he comes from an old family—he talks about his "900 year old name" in the poem. The Duchess is having her portrait painted by an Italian painter, Fra Pandolf. The monologue reveals, slowly, the Duke's almost pathological jealousy. It starts...
(The entire section contains 3 answers and 522 words.)