removing bold styling from part of a header

<h1 style="font-weight: normal;"></h1>

try this?

If you don't want a separate CSS file, you can use inline CSS:

<h1>This text should be bold, <span style="font-weight:normal">but this text should not</span></h1>

However, as Madara's comment suggests, you might want to consider putting the unbolded part in a different header, depending on the use case involved.

Yes you can add text inside <span> and override css. jsfiddle

html:

<h1>**This text should be bold**, <span>but this text should not</span><h1>

css:

span{
   font-weight: normal;
}

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.