Draw a single loop through the diagram, visiting every ball without intersecting itself. The loop makes a 90 degree turn in every black ball, but it goes straight in the cell where it enters and the cell where it leaves the cell with the black ball. It does not make turns in any of the white balls, but it does turn directly before and/or directly after visiting a white ball.
The black ball in the lower left corner borders to another cell in only two directions. Because the loop has to go through every ball and because right before and right after it goes through a black ball it has to go straight, we can draw lines from there two cells to the right and two cells up.
Through white balls a line has to go straight and on at least one side it has to make an angle. At the bottom row we see a white ball the line through which can only go horizontally and does not make a turn on the left side, so it has to go up in the cell to the right. Because there we encounter a white ball once more, the line can be drawn one cell up extra.
From the black ball in the center of the puzzle it is not possible to extend a line two cells down, nor two cells to the right, from which it follows that there are lines from that point extending two cells to the left and two cells up.
Through the white ball in the rightmost column there has to be a vertical line. On the top row it has to make a turn to the left. Everything can now trivially be connected.