Мне удалось решить это самостоятельно, не так уж и сложно. Это код, который я использовал:
object missing = System.Reflection.Missing.Value;
Word.Application wordApp = new Word.ApplicationClass();
Word.Document aDoc = null;
object readOnly = false;
object isVisible = false;
wordApp.Visible = false;
object filename = "d:\\Testdocs\\testfile.doc";
object saveAs = "d:\\Testdocs\\output.doc";
object oTemplate = "d:\\Testdocs\\Template.dotx";
aDoc = wordApp.Documents.Add(ref oTemplate, ref missing,
ref missing, ref missing);
aDoc = wordApp.Documents.Open(ref filename, ref missing,
ref readOnly, ref missing, ref missing,
ref missing, ref missing, ref missing,
ref missing, ref missing, ref missing,
ref isVisible, ref missing, ref missing,
ref missing, ref missing);
aDoc.Activate();
aDoc.set_AttachedTemplate(oTemplate);
aDoc.UpdateStyles();
aDoc.SaveAs(ref saveAs, ref missing, ref missing,
ref missing, ref missing, ref missing,
ref missing, ref missing, ref missing,
ref missing, ref missing, ref missing, ref missing);
aDoc.Close(ref missing, ref missing, ref missing);