That explains a lot, forgive the questions but the 'clients' on Edaboard range from school students to power station engineers so it was necessary to put a scale on the size of problem.
The relay solution will work fine but there are alternatives which are more efficient. As the trigger current is likely to be in the microamps region, a relay is really overkill.
If you need electrical isolation between the chime circuit and your control circuit, an opto-coupler will suffice and be much smaller. You connect it the same way as a relay but it has no moving parts, the only thing you have to be careful of is getting the polarity right as unlike a relay they are not tolerant of reverse connection and you have to limit the current to the input side. Typically they are a 6-pin or 8-pin plastic package, much smaller than a relay and run cold, taking only a fraction of the power used by the coil in a conventional relay.
If the chime and your control circuit do not need isolation and share a common ground connection (usually the negative side of the supply), you can still use the opto-coupler method but there is an even easier way, use a single transistor. What you are essentially trying to achieve is a voltage level conversion from the 5V to the trigger wire voltage which is quite simple to do.
Personally, I would use the opto-coupler as it works in all circumstances and is inexpensive and more versatile.
Brian.