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Here's a picture of a duplicated chromosome.
There are two chromatids, you see?
And they're held together by that yellow thing in the center.
It's called the "centromere."
The duplicated chromosomes are held together by a molecule called Cohesin. I've included a link to a web page with a good summary:
Cohesin is also involved in attachment of the mitotic spindle to the centromere. It appears that as chromosomes condense the chromatids are joined along their length by cohesin complexes. However by the time they line up at metaphase, the chromatids are only joined at the centromere. So my learned colleague @In YourHead is correct in saying that they are joined by the centromere but the cohesin complexes play an important part in regulating mitosis and meiosis too.