Vivienne,

My domain is accessible with and without the .www  

From your Weebly Dashboard, check your domain DNS records -

Under Domains - at the far right of your domain name are three dots ... 

Click the dots and click manage.  

Under DNS records you should see your A Records

My top 3 rows are:  

Type                    Host                      Points to

A Record             *                            199----------

A Record             @                          199----------

A Record            www                       199----------

Click edit if you need to add A Record(s).  

Hopefully this will work for you.

-- Anne