Open the file in GIF Construction Set Professional even though it isn't an animation you can treat it as an animation of one frame.
Double click on the image. You will get a dialog box called Edit Image.
In the Control Section, click the Transparent colour check box. Click on the eyedropper right under "colour". This will bring up the image in the middle of the screen. Point the eyedropper to the color that you want to make transparent and click on it. This will put a number in the box to the left of the eyedropper icon.
Click OK and the background color that you selected will probably show up as black. Save the file and then load it to your web page.
That should give you an image with a transparent background.