How to get the Tempered Sword
Shortly after you complete Gargoyle's Domain in the Dark World and you get the Titan's Mitt you'll be able to upgrade your sword to the Tempered Sword! To start with this quest you'll want to be in the Dark World. If you aren't then the fastest way to get there and to also be close to our destination is to take the Dark World portal north of Kakariko Village.
Once you arrive in the Dark World travel south through Kakariko Village (and south out of the southern exit of the village). After taking the southern exit you'll want to travel a little bit to the west and here you'll find a frog jumping behind some darker colored rocks.
Lift the rocks and speak to the frog that you find here and he'll start to follow you; this frog is actually the second gnome weapon smith! If you didn't run around Kakariko Village earlier in the game you may have no clue who this is and that's ok! With the frog following you head back to the Village of Outcasts and east 1 screen.
The house we are going to is the same place where we got the Magic Meter Upgrade earlier in the game. Once you head east 1 screen from the Village of the Outcasts use the Magic Mirror to come back to the Light World. Enter the house that is in the screenshot below and speak with the gnomes inside.
They will offer to temper your sword for a measly 10 Rupees which you should gladly accept as it'll increase the amount of damage you deal. To quickly finish the tempering return to Kakariko Village and then return to the gnomes and the sword will be finished. Congrats!
Before you leave this location altogether and go back to whatever it is you're doing - if you haven't gotten the Fourth Magic Bottle yet you may want to do so now! There is also a Heart Piece nearby that requires you to bang down some spread out purple stakes. Sadly I don't have a guide for heart pieces (yet!).
Additional Information: You'll need to Temper your sword to get the Golden Sword later on from the Fairy located at the Pyramid of Power.