If the chain exceeds functional limits, or is greater than 1.5% (0.36 inch per one foot), then it's time to replace the entire chain.
It's important not to connect a new section of chain to a section of worn chain as this may lead to it running roughly and is also likely to damage the drive.
It's also imperative to not continue to run a chain beyond 3% elongation. In these circumstances the chain will not engage the sprocket teeth properly which can lead to damage of chain, sprockets and possibly other machinery.
The most accurate way to measure chain wear is by using a tool such as FB's Professional Chain Wear Gauge.