S.D. Codified Laws § 15-14-13

If after the impaneling of a jury and before a verdict, a juror dies or becomes so sick as to be unable properly to perform his duty, or if a member of the immediate family of a juror dies or becomes so sick as to warrant the discharge of such juror, the court may order him to be discharged. In that case the trial may proceed with the other jurors if the parties so agree, or another juror may be sworn and the trial begin anew, or the jury may be discharged and a new jury then or afterward impaneled.