Multipart upload to Amazon S3 overwrites the last part

Question

I am doing a multipart upload with the .NET client library. The problem is that each part sent to S3 is overwriting the last part. Here's what I've got:

var fileTransferUtility = new TransferUtility(_s3Client);
var request = new TransferUtilityUploadRequest
    {
        BucketName = "mybucket",
        InputStream = stream,
        StorageClass = S3StorageClass.ReducedRedundancy,
        PartSize = stream.Length,//stream is 10,000 bytes at a time
        Key = fileName
    };

What am I doing wrong? Can someone guide me on this?