GZipStream compression not working

I'm trying to read in a file and compress it using GZipStream, like this:

using (var outStream = new MemoryStream())
{
    using (var fileStream = new FileStream(filename, FileMode.Open, FileAccess.Read))
    {
        using (var gzipStream = new GZipStream(outStream, CompressionMode.Compress))
        {
            fileStream.CopyTo(gzipStream);

            Debug.WriteLine(
                "Compressed from {0} to {1} bytes", 
                fileStream.Length, 
                outStream.Length);

            // "outStream" is utilised here (persisted to a NoSql database).
        }
    }
} 

The problem is that outStream.Length always shows 10 bytes. What am I doing wrong?

I've tried calling gzipStream.Close() after the fileStream.CopyTo line (as suggested in other forums) but this seems to close outStream too, so the subsequent code that uses it falls over.