The majority supported passing the bill in the Senate (in parliamentary jargon, they supported reading the bill for a third time). Since it has already passed in the House of Representatives, the bill will now become law.
The main idea of the bill is to recognise local governments in the Australian Constitution but it can't do this on its own. To amend the Constitution, there must be a successful referendum (as required by Section 128).
Rebellious Coalition senators
Coalition senators were split on this question. Seven Liberal senators and three National Party senators voted 'yes' while six Liberal senators and 1 National Party senator voted 'no'.
Liberal and National Party members are allowed to rebel (unlike Labor Party members), but it is increasingly uncommon.
What does recognising local governments mean?
Recognising local governments in the Constitution would mean that the federal government could directly fund local governments instead of having to fund them through state governments.
For an academic discussion of the effects of recognition, see Professor Anne Twomey's report (906 KB).
Background to the bill
See the bills digest (775 KB) for more background information.