No offense, but there's certainly a more elegant solution. I'm not sure why you're catching a specific exception just to throw a more general one either.

Code:
public string Encode(string str)
{
   byte[] encbuff = System.Text.Encoding.UTF8.GetBytes(str);
   return Convert.ToBase64String(encbuff);
}
public string Decode(string str)
{
   byte[] decbuff = Convert.FromBase64String(str);
   return System.Text.Encoding.UTF8.GetString(decbuff);
}