You can clone it to get the entire contents, like this:
var html = $("<div />").append($("#div1").clone()).html();
Or make it a plugin, most tend to call this "outerHTML", like this:
jQuery.fn.outerHTML = function() {
return jQuery('<div />').append(this.eq(0).clone()).html();
};
Then you can just call:
var html = $("#div1").outerHTML();