You could wrap the not-bold text into a span and give the span the following properties:
.notbold{
font-weight:normal
}?
and
<h1>**This text should be bold**, <span class='notbold'>but this text should not</span></h1>
See: http://jsfiddle.net/MRcpa/1/
Use <span>
when you want to change the style of elements without placing them in a new block-level element in the document.