A wheat cultivar designated HY 319-SWW is disclosed. The invention relates to the seeds and plants of wheat cultivar HY 319-SWW, and to methods for producing wheat seeds and plants by crossing wheat cultivar HY 319-SWW with itself or another wheat cultivar or wheat plant not designated a cultivar. The invention also relates to methods for producing seeds and plants of wheat cultivar HY 319-SWW containing in its genetic material one or more transgenes and to the transgenic wheat plants and plant parts produced by those methods. The invention also relates to methods for producing seeds and plants by mutagenesis of wheat cultivar HY 319-SWW. The invention also relates to hybrid wheat seeds and plants produced by crossing wheat cultivar HY 319-SWW with another wheat cultivar.