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Make sure it is the tank that is leaking. Sometimes a hang-on power filter is overflowing or leaking, or an airstone is spraying water out the back corner, or a piece of paper, cloth or tubing is dangling into the water and "wicking" water out.
Empty the water - and everything else - from the tank. The sealant will need to applied to the inside of the aquarium, where it will be pushed into the seam by water pressure.
Remove old sealant with a razor blade scraper. If the leak is apparent on a side seam, remove all the sealant from the entire length of seam. If water appears mysteriously at the tank's bottom, scrape out all four bottom seams.
Prepare the seam. Some aquarists use rubbing alcohol to remove oily fingerprints, others simply wipe with a damp rag. It is essential that the area be extremely clean and completely dry.
Buy silicone sealer. It doesn't matter if you get it at the local fish store or hardware store, but make sure the packaging plainly indicates that the product is safe for aquarium use. Some products contain toxic mildew-retarding chemicals that will kill fish.
Apply sealer. Hold the tube at an angle and push it forward while squeezing out the sealer, forcing sealer into gaps. Some hobbyists then run their fingers down the bead, further pushing the sealer, but sealer is extremely difficult to remove from skin, so this is not recommended. If you feel a need to manipulate a wet seal, lay clean, dry plastic bags over the seals first. Leave bags in place until sealer is dry; they will then peel off easily.
Allow to cure. Follow sealant manufacturer's directions as to how long to wait before re-filling tank. Adding water too soon will weaken the seal and possibly contaminate the water.