The Greek of the phrase in 1 Thess 4:16 is φωνῇ ἀρχαγγέλου = "voice/sound of archangel". There is no definite article in the Greek, but that is not to suggest that there should not be one in English. For example, note the correct translation of John 1:1 -
Ἐν ἀρχῇ ἦν ὁ Λόγος = in the beginning was the Word
Note that there is no definite article in the Greek but there is need of one in English for reasons that need not concern us here. This is because the use of the definite article and the grammar rules associated with the definite article are different in the two languages. Latin (from which DRB was produced) has no articles, not even a definite article. It is regular source of debate as to whether one needs to be supplied in English or not when translating from Latin (most times it can be worked out).
Back to 1 Thess 4:16. The need or otherwise of a definite article before "archangel" is debateable here. The word archangel only occurs twice in the NT: here and in Jude 9. In the Hebrew it also occurs in Dan 12:1 and simply means the leader of the angels. Such a being is listed in Josh 5:14. Ellicott observes on this text that:
That the Lord's host must mean the angels is clear from such passages
as Genesis 32:2; 1 Kings 22:19; Psalm 103:20, 21; Psalm 148:2; St.
Luke 2:13 (aft 2 Kings 6:17). Hengstenberg, in his 'Christology,'
illustrates by Matthew 26:53. Two opinions have been held by the early
Church concerning this manifestation. The first regards it as the
appearance of the Son of God in a visible form; the second supposes it
to have been a created being - an angel - through whom Jehovah was
pleased to manifest Himself. The former opinion was general in the
earliest ages of the Church.
If that conclusion is correct, then "the Archangel" is appropriate. However, if there are numerous archangels (we do not know this), then "an archangel" is appropriate in 1 Thess 4:16.