What does "devil a bit" mean?

I’ve read this in older books, and I get the impression that it means “not at all”, but the construction doesn’t make sense. Am I right as to the meaning? And how should I interpret the form?

Here’s one usage:

And that’s the reason why, if a man speaks his mind too freely against the gospel according to Saint Mat Thew, the Devil-a-bit does Mat care about it.

It seems that this is a slang saying from Dublin, Ireland. It means "nothing at all". You may read the whole thread for the relevant discussion.

Eric Partridge has "devil a bit" in his Dictionary of Slang and says:

devil a bit (says Punch), the
A firm though jocular negative: colloquial: circa 1850-1910. Without says Punch it goes back to earlyish 17th century: Pepys uses it thus on 3 April 1668.

"Divil the bit" & less commonly “divil a bit" is an Irish English idiom meaning none at all.

"Devil a bit" sounds like an anglicised version. Punch is a reference for Irish English the way Tom & Jerry is a reference for African-American English.

J.M. Synge used "the divil a one" in The Playboy of the Western World to mean "none at all". You can also hear it in George M. Cohan's song "Harrigan": "divil a man can say a word agin me", clearly meaning in context that no one can disparage him.

'Divil' can also be used in other negations. An example in song would be As I roved out / The night visit (same song, two names) — which uses 'Divil the one'/'Divil the man' to mean no-one.

And will you come to me mammy's house
When the moon is shining clearly.
And will you come to me mammy's house
When the moon is shining clearly.
I'll open the door and I'll let you in
And divil the one will hear us.

So I went to her house in the middle of the night
When the moon was shining clarely.
So I went to her house in the middle of the night
When the moon was shining clarely.
She opened the door and she let me in
And divil the one did hear us.

There we lay till the break of the day
And divil the one did hear us
There we lay till the break of the day
And divil the one did hear us
She arose and put on her clothes
Saying "Darling, you must leave me."

It's a traditional song, not sure when it was written. Just to provide other contexts. I'd agree with others that it's still used in Ireland, but seen as archaic, so may be slightly tongue-in-cheek if people do use it.