Expat Health Insurance
Cigna Global Health Insurance.
Medical insurance specifically designed for expats. With Cigna, you won't have to rely on foreign public health care systems, which may not meet your needs. Cigna allows you to speak to a doctor on demand, for consultations or instant advice, wherever you are in the world. They also offer full cancer care across all levels of cover, and settle the cost of treatments directly with the provider.
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Under what circumstances are you moving to France - retirement, employement, etc...If it's the former you'll need to complete an E121 form to access healthcare, if it's the latter a certain percentage of your income will be docked accordingly.
As a British expat, it makes the most sense to take out the CMU scheme, which comes with high-premiums attached, but which covers a lot. I'm not sure if it covers monthly blood tests, but have a look at the CMU site (you may need to use Google Translate to turn it into English) and see what it says.
Regardless though, bring enough of your regular medication to last you a few months in France while you figure out the best way to obtain it.