Pattern brush alignment for a circle.

Here are my two attempts at this simple pattern:
The example on the left clearly shows the alignment that I want to correct.
Is there a way to adjust this alignment? Playing with the scale and fit options (of the pattern brush) didn't seem to help.
Would you suggest another method for creating this circular border?
Thanks!
Windows 7 | Ai CS5

Thanks all.
scott w:
Sigh, I forgot all about it needing to fit inside of a rectangle.
I looked at some of my old patterns
and remembered that I was making the lines with the rectangle tool instead of the pen tool,
so that the pattern's "lines" could be aligned to the inside of grid lines.
Using this method doesn't seem to produced any alignment gaps.
Now that you have introduced the idea of using a defining rectangle,
I made two versions. One using my original rectangle tool / grid method,
and the other using a defining rectangle and the pen tool for the lines.
Rectangle tool method:
Rectangle tool method result:
As you can seen it slightly expands the rectangles, thereby eliminating the gap.
However, these "lines" don't look as clean as the alternative method.
Defining rectangle (invisible) / Pen tool:
Result:
Now we could use 'flip across' in the pattern options to eliminate the gaps,
but this produces a slight misalignment, which only the nuttiest of nit-pickers would even notice:
Unfortunately, I appear to be one of these nit-pickers.
So we are back to your method of the protruding path.
I need some details.
It seems like you didn't use the grid at all.
Did you create the defining rectangle first?
How did you align the path to the inside of the rectangle?
How did you determine the necessary protrusion length on each side?
Thanks!

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