The tune is presumably 'traditional' and named after the beautiful village Bunessan on the Isle of Mull, Scotland. [good chippy]. As previous contibuters have said, there are several hymns set to its tune, including a setting of the odd verse in 'St Patrick's Breastplate' which goes to a different tune to the rest of the hymn. James Quinn, Roman Catholic hymnwriter, took those words and made a new hymn of it.
So Paddy's vest, as mentioned above, more correctly known as St Patrick's Breastplate and starts with 'I bind unto myself today....'
the odd verse within it being 'Christ be with me, Christ within me..'
James Quin's version of the odd verse, set to Bunessan goes 'Christ be beside me,
full text below

Christ be beside me, Christ be before me,
Christ be behind me. King of my heart.
Christ be within me, Christ be below me,
Christ be above me, never to part.

Christ on my right hand, Christ on my left hand,
Christ all around me, shield in the strife.
Christ in my sleeping, Christ in my sitting,
Christ in my rising, light of my life.

Christ be in all hearts thinking about me,
Christ be in all tongues telling of me.
Christ be the vision in eyes that see me;
in ears that hear me, Christ ever be.

Words:        adapted from St. Patrick's Breastplate by James Quinn
Tune:        Bunessan [Morning has broken]

nice version I'm sure you'll agree.
Ask me about the day job.

Geoff