To create a custom Weebly theme, first go to the Design tab of the editor and choose one of Weebly’s available themes. Then, back in the Design tab, click on the “Edit HTML/CSS” button near the bottom of the sidebar:

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This will open the editor. Press save in the upper-righthand corner and it will ask you for a name for your new custom theme. It may be necessary to make a change in one of the theme files for this to come up (just add a space somewhere in the CSS or something):

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Name your theme whatever you want and your website will be set to use your custom theme (as opposed to the Weebly theme you chose in the beginning). You can now throw away all their HTML and CSS and replace it with code that is entirely your own.