Recently, I read a 2012 document from the European Commission about the Digital Single Market (A coherent framework for building trust in the Digital Single Market for e-commerce and online services).
In the report they confirm the increase of cross-border e-commerce in Europe and they look at trends and obstacles.
They also examine the potential benefits for customers and real opportunities for businesses to reach more consumers to grow.
Here is what I discovered.
Unfounded fears and an untapped market online The 66-pages report is built through a consumer market study in 12 European countries and complemented by other studies, Eurostat and industry data.
The points I've picked up for you:

• Many obstacles to e-commerce are common to domestic and cross-border e-commerce
• Information, redress and enforcement are the 3 areas where to act to boost a cross-border e-commerce strategy and to increase consumer trust and confidence
• Many fears are unfounded both for retailers and consumers
• The level of confidence of consumers who actually buy online cross-border is high
• Geographic location and knowledge of the language in which offers are made are important factors underpinning cross-border flows of trade
• The UK and France are 2 of the 3 most mature e-commerce economies in Europe

All are encouraging facts: there is a big market behind the screen and for the companies prepared to make the effort the rewards can be remarkable.
What is your British product about? Is it famous because it is greener? Because it has the best value or the most advanced technology? Or because it is Made-in-Britain and leader in the sector? Do you sell a British product that French customers will not find anywhere else? Do you rely solely on your British online shop? Customers' behaviour is changing: they are aware, they search and they compare.
They spend their money more carefully.
Buying in shops remains a real pleasure and a social experience for many French people.
For others attracted by e-commerce, you can help them to feel confident on your British website.
A checklist to increase trust in your e-commerce website The EU report shows clearly that real problems when shopping online - in your own country or cross-border - are related to customer service (and especially with electronic goods): delivery, non-conformity and Redress.
Simple things can be put in place to:

• Save your time
• Get less return from disappointed French customers
• And be compliant with the EU consumer legislation

Have a look here at the 10 golden rules given to customers shopping online abroad.
By addressing these 10 rules you can boost French customers' confidence in your website.
Keep in mind that the more information you can give them in French the better.
Let people know in French about:

• Your payment terms and conditions
• Your price per unit, kilo, litre or meter, etc.
  if relevant
• Some details about the components, material, ingredients you use and where you source them (Made-in-Britain?)
• A link to a currency website so they can check how much they pay in euros (some websites display historical rate tables so it is easy to know the rate at a precise date)
• The time needed for payments transfers between the UK and France
• The postal / courier delays between the UK and France

It is also helpful to tell them:

• How they will be answered if they call you.
  In English or in French?
• How you will keep in touch.
  Do you invite them to register to your newsletter? Can they follow you on social media in French?
• And if you have other products or offer for them

Have you already a French page on your website? If so, leave it clear and nice.
Design a simple PDF file where you can display as much information as you want in French.
It is easy to update.
Check your French copywriting.
And make the PDF file easy to find, to download and print.
A lot is happening in Europe to help businesses and consumers with cross-border e-commerce.
Keep updated with e-commerce European policies.
You will find online a lot of resources from trade and professional bodies.