You need to be Adult Link, have freed King Zora, and have unlocked Epona to do this quest. It also helps to have the Sun's Song, though it is not essential.
First, go and speak to the Cucco Lady in Kakariko Village. Take the Pocket Egg from her, before playing the Sun's Song to move the game forward to the next day. The egg will then hatch into a Pocket Cucco.
With the Pocket Cucco, go and speak to Talon (he is sleeping in Kakariko Village - use the Pocket Cucco to wake him). Return the Cucco to the Cucco Lady, who will then give you Cojiro, another Cucco, to pass on to her brother.
You now need to head into the Lost Woods. Turn left once inside. Give the Cucco to the blue figure on the stump. In return, he gives you an Odd Mushroom, asking you to deliver it to the witch doctor in Kakariko.
A timer then begins, giving you 3 minutes to return to Kakariko and pass on the mushroom before it goes off. You can find the witch doctor behind the potion shop.
The witch doctor will make you an Odd Potion from the Odd Mushroom (makes sense...). Go back to the Lost Woods, turning left again. There will be a girl in the blue man's place. Give her the potion and she will give you the Poacher's Saw.
Head to the canyon in Gerudo's Valley. Cross the bridge, if it is in place, or use Epona or the Longshot to get across the gap. Speak to the carpenter in his tent. Give him the Saw and he will give you the Broken Goron's Sword.
You then need to go to the top of the mountain using the Bolero of Fire. Move round the right side of the cliff. Speak to Biggoron, who should come to find you here. He agrees to fix the sword but only in return for Eyedrops.
Take his Prescription and go to Zora's Domain. Show the Prescription to Zora, who gives you the ingredients for the Eyedrops. Take the Eyeball Frog to the scientist by the lake at Hylia. Again, you only have 3 minutes to get there before the frog goes bad. The Serenade of Water also makes it spoil, as that would be cheating!
Give the scientist the frog. Once you have the Eyedrops, rush back to Biggoron. The Eyedrops (for some reason) only last 4 minutes, so go as fast as you can. The Bolero of Fire spoils the Eyedrops as well, so you have to go the old fashioned way.
Biggoron agrees to produce a new sword for you. After 3 days, you can return to him with his Claim Check to get the sword. Use the Sun's Song if you are too impatient to wait!
Congratulations, you now have the incredible Biggoron's Sword!
Added by: MongooseGeneral Jun 22nd 2011, ID#62