I think it should be read double. It’s possible to just say tha mi an Dùn Dè (as there is no definite article before Dùn Dè, an Dùn Dè can only mean in Dundee and thus an doesn’t need to be doubled), but that also would need to be written without doubling the preposition. If the speaker does say an Dùn Dè instead of ann an here, maybe that’s a mistake in recording?
See eg. this part of Akerbeltz ann an article:
And lastly, before place names an also shows up relatively frequently, as in bha iad an Glaschu "they were in Glasgow". Basically, an is sufficient because no potential confusion will arise from that sentence. You can not come up with a meaningful sentence putting an "their" or an "the" in front of Glaschu - without sounding weird or wrong.