Oui! I got a high five from the sweet (marginally bilingual) girl at the déppaneur I go to most often when I was able to tell her the total for my gas in French the first time, and then said, "J'apprends, lentement [and then she finished the sentence with me] mais sûrement!" :)
« Dépanneur » (not "déppaneur") means "corner shop/store", i.e., a convenience store. It is Canadian French.
It is not an idiom (an expression that has a different meaning than the literal translation). It is a common expression in English, however.
Sûrement means certainly, but I should have guessed that it's different in this case or the answer is wrong?!?
Surely and certainly are synonyms, so if it means one it is quite likely to mean the other too, though both translations may not be correct in every context - here "certainly" wouldn't be correct, because it is a fixed expression. I usually translate "sûrement" as "surely" and I guess the fact I'm sometimes (not always!) marked wrong is because (as far as I know) English prefers "certainly" to "surely" more often than French (I don't remember how is it spelled - "certainement"?).
The mills of God grind slowly but surely. Is there any language unfamiliar with this proverb?
The biggest problem is that « mais » doesn't mean "and". It means "but". « sûrement » doesn't exactly mean "carefully", but more like "certainly". And lastly, "Slowly but surely" is an idiom and a fixed expression, so it is the best translation.
Because "slowly but surely" is a fixed idiom and as such there's only one way to say it.
Is there supposed to be an audible liaison between mais and surement, or is that an artifact of the recording?
A liaison is a phenomenon whereby a normally silent consonant at the end of a word is pronounced at the beginning of the word that follows it. This typically occurs when the next word begins with a vowel or vowel sound. As you can see, that doesn't apply here.