You can't remove the commas from the existing string - although you could create a different string before splitting. That's typically what you do with string processing - create a whole sequence of strings, performing replacements, trimming etc along the way.
A simpler solution is probably to use StringSplitOptions.RemoveEmptyEntries though:
string[] parts = str.Split(',', StringSplitOptions.RemoveEmptyEntries);
array = Array.ConvertAll(parts, int.Parse);
(Note that the String.Split(char, StringSplitOptions) overload is not available in the desktop framework; you'd need to call String.Split(char[], StringSplitOptions) - just change ',' to new[] { ',' } in the calling code.)
You could potentially include StringSplitOptions.TrimEntries too, so that "1, 2 , 3 ,4" would be handled equivalently to "1,2,3,4".
This is assuming you just want to handle users adding more commas than expected. It won't help if the user enters "1,2,xyz" for example.