Replacing a copper downspout is a difficult job for a do it yourselfer and you should only attempt it if you are very familiar with working with copper and have a lot of patience. There are alternatives to using copper if you think this task may not be something you want to try.
Think About the Alternatives
Copper downspouts aren't used on homes much any more due to the cost, the likelihood of them being stolen and the skill it takes to install them. Copper gutters and downspouts cost around $20 and up per foot in most areas. Copper is a very expensive metal and has a lot of appeal to thieves for its resale value at scrap yards. You may notice older buildings such as schools that have copper downspouts, gutters and connectors only on the levels above first floor. Most likely the easy to reach pieces were stolen.
Copper is difficult to install as you must wear clean gloves since body oils will mark the copper and cause it to corrode and leave your prints on it forever. Joining copper is also a skill that not everyone has. Most professionals use an iron rather than a soldering tool. You will most likely have to get the copper downspout made by a professional as this isn't an item you can buy in a home improvement store. Most stores won't even special order them. You might want to consider galvanized metal and paint it copper color or the new galvanzied with a copper coating on it to save money.
Step 1 - Measuring
You will need to observe the old adage 'measure twice, cut once' when working with or ordering copper downspouts due to the cost. This isn't an item you want to purchase twice. If you have the old downspout, use it for measurement. If you don't have it, then you need to measure the distance from the opening in the gutter where the downspout will be inserted to the ground level. Add 2 inches for overlap into the guttering. If you are ordering the downspout the supplier may want to do the measuring for you.
Step 2 - Clean
Once you have the downspout at the proper measurement you will need to clean the end to be soldered into the guttering so it will accept the flux. Do this with a fine grit sandpaper. Then you will need to put flux on the areas to be joined using your small brush. Don't use too much. The flux allows the solder to stick to the copper.
Step 3 - Solder
Solder the 2 pieces together using your torch. You will want to let the solder flow into the joint naturally as it melts and becomes liquid. Be careful to not use too much. Solder is hot, and can burn you very badly so be careful with this step. Once the solder is set, wipe with a clean cloth. Do this with all the joints of the downspout.