Thanks for all of the help. I ended up doing something like this:
If File.Exists(lstrFileName) Then
' Get the one line that has AS OF DATE
For Each line As String In File.ReadAllLines(lstrFileName)
If line.Contains("AS OF DATE") Then
lstrFileContent = line
End If
Next line
'Look inside the string for the date
Dim lintStart As Integer = InStr(lstrFileContent, "DATE")
'MsgBox("lintStart " & lintStart)
Dim lstrFileAsOfDate As String = lstrFileContent.Substring((lintStart + 11), 8)
Dts.Variables("strFileAsOfDate").Value = lstrFileAsOfDate
'MsgBox("datFileAsOfDate " & Dts.Variables("datFileAsOfDate").Value)
Dts.Variables("blnContinue").Value = True
Else
Dts.Variables("blnContinue").Value = False
End If