If you already have an existing blog, you will find that there is no easy solution to transfer everything to your "new site" (nor would you want to). If you've been using Blogger for a while now, you would have gotten accustomed to many options and quirks that are just not available yet on Weebly's own simple blog solution.
Here is a tutorial of how to embed your existing blog from Blogger right into your Weebly site:
1. Log in to your Weebly account and select your site.
2. Click on the "Pages" menu and choose "Add Page"
3. Select "Standard Page" and name it Blog (or whatever you like)
4. Click "Save & Edit".
Now to make sure your blog is the right size:
5. Log in to your Blogger account and select your blog
6. Click on "Template" and then on "Customize"
7. Click on "Adjust widths", choose what suits your needs and click "Apply to Blog" when finished.
Here are the settings you need to play with in order to make your blog fit inside whatever website template you are using. Back to Weebly:
8. Scroll down on the left sidebar and choose "Embed Code" from the menu.
9. Put the Embed Code to your page and click on it.
10. You will need to copy and paste this code into it:
<iframe allowtransparency="true" scrolling="yes" frameborder="0" width="1280" height="1280" src="http://getvladimir.blogspot.com" style="margin: -140px 0 0 -195px;" > Put name of your blog here </iframe>
11. Change the HTTP:// address to your blog address and enter the name of the blog.
12. Ajdust the "margin: -140px 0 0 -195px;" to make sure your blog fits.
The first "-140px" means that Weebly will show your blog starting from 140px from the top. This effectively is like you cut 140px from the top. The last "-195px" tells how much you want to cut from the left side.
13. Try it a few times and adjust the margins a little bit until you get it right.
That's it! Now you have your blog on your Weebly site without the need to transfer or synchronize.
P.S. This could be used to implement pretty much any content into your site.