Change Namespace of element and children

Hi All, new to the list but not new to XSLT.

I want to change the namespace of an element using XSLT. My problem is,
I can change the namepsace of a single element easily by doing a
template match and then simply re-adding the element and specifying the
new namespace ... like so:

   <xsl:template match="MyElement">
      <xsl:element name="MyElement" namespace="http://www.mystuff.com">
         <xsl:apply-templates select="@*|*|text()"/>
      </xsl:element>
   </xsl:template>

But, the problem is, child elements of MyElement end up with the default
(empty) namespace ... like so:

<MyElement xmlns=http://www.mystuff.com">
    <MySubElement1 xmlns="">12345</MySubElement1>
    <MySubElement2 xmlns="">12345</MySubElement2>
</MyELement>


Any ideas?

Thanks,

- Kevin