An apparatus for grinding a concrete floor surface, in order to perform grinding with a manual, a pullable or a ride-on concrete trowel machine is provided. The apparatus comprises a concrete trowel organ to be used with the concrete trowel machine and a diamond grinding arrangement, such as one or several diamond disks or like, to be coupled in connection with the concrete trowel organ removably by means of a fastening arrangement. The fastening arrangement for coupling the diamond grinding arrangement in connection with the concrete trowel organ comprises a fastening frame, being in connection with the diamond grinding arrangement that exits on a scrubbing surface of the concrete trowel organ, and a fastening part extending from the fastening frame and, while going through the concrete trowel organ, getting supported against a back surface of the same.