Every time I see this expression, I can't help thinking it's grammatically wrong. Is it grammatically acceptable? Why is it used extensively in this form?
The phrase is quite old: it was part of the Book of Common Prayer from 1662 (see http://www.eskimo.com/~lhowell/bcp1662/occasion/marriage.html). (For all I know, it could be older even than that.) But fixing it as the official language of a ceremony cemented the phrase in that form, even as the language changed around it. It's probably best viewed as idiomatic; you wouldn't want to say something else using the same form without a very good reason, but that particular phrase is a widely recognized feature of the language.
(Note, by the way, that the form in the BCP really was with "till" rather than "until"; this isn't surprising, since "till" is actually the older of the two words.)
All of the answers omit a key point, or at best imply it. While "death" here is the subject, most people take "death" to be a temporal indication. This is why they expect the pronoun to be "we": I promise to stay with you till (the time of) death. Of course, it really means "until death separates us."
Both (1) till death do us part and (2) till death us do part are found if you ngram-view them, but not (3) till death do part us, which is surprising since it would be the normal word order:
till (conjunction) death (subject) do part (present subjunctive of 'do' + bare infinitive of 'part', emphatic form) us [from each other] (direct object of the verb 'part')
Structures parallel to these would be:
(1) as long as | you | do | me | love (?!)
(2) as long as | you | me | do love (?!)
(3) as long as | you | do love | me