No, the system will not automatically write to the 1TB HDD if the 300GB HDD fills up. You have to instruct Windows where you want things. Moving things after the fact is not the best solution, but it can be made to work.
How you configure your system will depend on what you're doing. What exactly are you doing that is constantly writing so much data to your 300GB drive it is going to fill up? If it's something such as recording TV using a TV tuner card, that could be instructed to use your secondary drive, or if you're downloading things from the internet, save them to your secondary instead of your primary HDD. There must be something you are doing to use the space, but without knowing, I can't suggest much more than I have.