Yes, the 1km radius rule (and one hour time limit) only applies to outings that are purely for exercise.
Generally, you can only go to shops for essential goods (or other acceptable purposes) as stated on the attestation exemption forms, so if you are going shopping for food or other essential items, that will be allowed.
If you are cycling to the supermarket - or other food or essential shop or service that is more than 1km away - and back, you will be allowed to do so for that purpose. The purpose must be stated on the form.
You can also use your bike for transport, for any other acceptable purpose given on the form.
You will need an attestation exemption form, which is available to print and download here (in French). There is an English version available and also an easy-read version with pictograms. There is also a digital version available via the TousAntiCovid smartphone app.
You will need to tick the second box on the attestation form, which says “des achats de première nécessité” (necessary purchases) for food and basic necessities, or the box that says "pour faire mes courses". This permits you to be away from home for more than an hour and travel beyond the 1km limit.
Most of the major supermarkets and many other stores have put in place click-and-collect options, allowing you to order groceries online and arrange to pick them up at a certain time to limit contact and reduce the spread of Covid-19.
Non-essential items will not be available at the supermarket, as announced by Prime Minister Jean Castex today with more details set to be announced tomorrow.
Cycling purely for exercise must be done within 1km radius of your home and within one hour (the sixth box down on the attestation form).