I moved into my house about a year ago and have never really paid much attention to the fireplace because it has never been cold enough to use it (I live in Georgia and we can usually get by with the central heat/air). Now it is 25 degrees outside and I wanted to build a fire this evening, but after a quick inspection I noticed a couple of problems.

First of all, the chimney is nasty and needs a good cleaning. Second, there is a fairly large gap (about 1/2") between the firebox and the hearth. The hearth is covered in ceramic tile and the builder used grout to fill the 1/2" gap. Now that it has been in there for a while it has started to crack and chip out, exposing the wood framing underneath.

What is the best way to fill this gap back up? I have heard a few different things ranging from filling the gap with bits of broken tile and re-grouting it, to simply filling it with a high temperature caulk. I want to do this right the first time so that there is no risk of embers falling down in the crack and setting fire to the wood underneath.