adverb
swear by God; absolutely
Used to emphasize the truth or seriousness of a statement, often in oaths or strong assertions. Can feel slightly dramatic or old-fashioned.
神掛けて嘘は言っていない。
I swear to God I'm not lying.
神掛けて、その約束は守ります。
I absolutely promise to keep that promise.
誓って is a more common and neutral way to say 'I swear' or 'I vow', while 神掛けて explicitly invokes God and carries a stronger, more dramatic tone.
From 神 (kami, 'god') + 掛けて (kakete, te-form of 掛ける, here meaning 'to stake' or 'to swear upon'). Literally 'staking on God', i.e., swearing by God.