This is a bit of a hack, but might be suitable for small amounts of text. If you aren't worried about where the line breaks come in the block of text, it could be simplified, but it looks to me like you wanted specific line breaks and for all lines to be justified.
It works by making sure there's some invisible text (the non-breaking spaces) that's forced onto the next line as a unit (that's the non-breaking bit), but that won't go onto a second line (as it's white space).
By forcing the line height to be smaller than 100%, we make the spacing between the lines adjust as we want.
By limit the width of the paragraph block, we prevent unfortunate wrapping or non-wrapping.
I suspect it'll be possible to centre the text block as well, perhaps by wrapping the paragraph in another div.
Code:
<html>
<body>
<p style="line-height:60%; text-align:justify; min-width: 20em; max-width:40em">
This is some text in a paragraph to be <br>
fully justified by having lots of spaces at the <br>
end which will cause the last line to justify. </p>
</body>
</html>