How would the end of the first paragraph be marked ??
if it was by a <br> then just do
$article = explode("<BR>", $text);
$first_paragraph = $article[0];
if the paragraph is defined as being of a certain length then you can use the substr() function.
So if it was the first 50 characters you would do
$first_paragraph = substr($article, 0,50);
To do a similar substring with mysql you could use
$sql = "select SUBSTRING_INDEX(field, ' ', 50) FROM tablename"; //this would return the 1st 50 words from the field