Necklaces with chain links under two millimeters long/wide have a certain refinement to them. They are simple statements to accessorize a more flamboyant outfit. However, they are the one piece of jewelry that breaks the most often. You can take your fine-link necklaces to a jeweler for jewelry repair, but if the necklaces are just costume jewelry, you may pay more for the repair than what you did for the necklace. That said, here is how you can DIY your fine-link, costume jewelry necklaces:

Get Some Sort of Magnifying Device

You will need a magnifying glass, a microscope, or some other type of magnifying lens device. This will help you see the links in your necklace better as you work on it. There are also magnifying monocles and magnifying eyeglasses which can help and may be a better hands-free option.

Get Tiny Tools

Tiny screwdrivers, tiny forceps, and tiny pliers often used for tying fishing lures are exactly what you need. You will have to force open a link on the broken end of the chain. Then use the forceps to help link the open chain to the link on the opposite end of the necklace. The tiny pliers help close the link you opened to reconnect the disconnected links. 

You could also buy jewelry crafter's tools. These tools are meant for this kind of project. You can buy them from wherever bead-crafting and jewelry making supplies are sold.

Buy Extra Links

If you go looking for jewelry crafter's tools, buy some extra necklace links while you are there. These are often sold by the hundreds in various finishes and colors. The spare links will be helpful to you if you accidentally break another link on your necklace or need to fix a different necklace altogether.

Have a Stable, Flat Surface on Which to Work

A kitchen table, a sewing table, a desk; all of these are good surfaces for which to repair your necklace. If anything drops off the necklace it lands on the table and not on the floor or your lap where a small link cannot be seen. It also helps stabilize your arms for this tiny detail work, thus preventing any further damage to your necklace.

Have Patience

Even a jeweler takes his/her time to repair expensive chain-link necklaces in the shop. You want to do the job right, not quickly. It may take some practice at first, but you will get the hang of it and then your favorite costume necklace will be fixed and ready to wear again.