This is a question of style, and styles have changed over time. At one time it was fairly common to use two periods in a situation such as the example above, but people found that ugly. Then there was a period of time (I'm thinking ca 1960) when the period would never fall before the closing quote, even if the quote was a "complete sentence".
Current style, when a quote comes at the end of the containing sentence, is to place the period before the closing quote if the quoted text is a complete sentence, and place it after the quote otherwise. However, note that if the following sentence begins, eg, with an acronym or proper name then using this policy could result in ambiguity for the reader, and hence the second period might be justified. So to some extent the choice is at the writer's discretion.