Paul. I appreciate your help. The problem is solved. Here is what I did. I added Application.EnableEvents = False & Application.EnableEvents = True.
Here is the code:

Private Sub Worksheet_Change(ByVal Target As Range)

 'This code is activated if the delete key is activated,
If Trim(Target.Value) = Empty Then
Dim Ans As Integer
Application.Speech.Speak "Two appointments were scheduled on your calendar previously. One for a Contact Letter, one to Cancel the Consult ", SpeakAsync:=True
       Application.Wait (Now + TimeValue("00:00:2"))
     Ans = MsgBox("Two appointments were scheduled on your calendar previously. One for a Contact Letter, one to Cancel the Consult " & vbCrLf & vbNewLine & "Click Yes to delete the future appointments,if veteran contacted you.  Click No, if there was no contact from the veteran.", vbYesNo + vbInformation, _
                       "Vocational Services Reminder")
      Select Case Ans
          Case vbYes
       '[code if Ans is Yes]...
       'Opens Outlook appointment Calendar.
Dim olApp2 As Object ' Outlook.Application
Set olApp2 = CreateObject("Outlook.Application")
olApp2.Session.GetDefaultFolder(olFolderCalendar).Display
Application.EnableEvents = False
Range("$M$3:$M$329").ClearContents
Application.EnableEvents = True
  End Select
  Select Case Ans
   Case vbNo
  '       ...[code if Ans is No]...
 MsgBox " Enter the reason in the Column. You can choose from the drop down list or enter a new one.", vbInformation, "Vocational Services Database - " & ActiveSheet.Name
  ActiveCell.Offset(0, -1).Select
      End Select
      Exit Sub
End If
I placed Application.EnableEvents = False, before Range("$M$3:$M$329").ClearContents, & Application.EnableEvents = True, after. I think it was caught in a loop without the False statement. The cell started out filled, then deleted. If "Yes" was selected it went back to code line #1 If Trim(Target.Value) = Empty Then. By adding both the False & TRUE statements, I was able to prevent the loop & turn the Excel Events back on for future use.