Advocates for a second Brexit referendum demonstrate outside Westminster on Wednesday. (Alastair Grant/AP)

With just 30 days to go until Britain is scheduled to leave the European Union, Parliament on Wednesday endorsed Prime Minister Theresa May’s concession that if the lawmakers cannot agree on a deal to sever ties with the continent, Brexit might have to be delayed.

But beyond a consensus about possible postponement, the gridlock that has seized the British political class continued.

Last month, lawmakers overwhelmingly rejected May’s 585-page withdrawal agreement, negotiated over two years with her European counterparts. May hasn’t yet managed to sweeten the deal.

On Wednesday, Parliament decisively voted against a one-page outline of a Brexit plan proposed by the opposition Labour Party and its leader, Jeremy Corbyn.

Labour’s vision for a soft Brexit would have seen Britain remain closely aligned with E.U. customs, tariff and regulatory regimes and the continent’s single market. Such a relationship would have meant that Britain would continue to allow E.U. migrants to live and work in the United Kingdom, while withdrawing from the E.U. legislature.

The British Parliament’s rejection of Labour’s plan — widely expected — boxes a reluctant Corbyn into throwing his weight behind a new public vote on Brexit.

Amid fears of further defections from within its ranks, the Labour Party announced earlier this week that it would support a second referendum if its own Brexit proposal was rejected.

Corbyn confirmed in a statement Wednesday night: “We will back a public vote in order to prevent a damaging Tory Brexit or a disastrous no-deal outcome.”

While momentum for a second referendum has grown in recent months — and advocates swarmed Westminster on Wednesday — it’s not clear how many lawmakers actually want a do-over.

A second referendum would enrage parts of the British public, including many of the pro-Brexit demonstrators outside of Parliament on Wednesday, some holding aloft placards that read “leave means leave” and “Brexit means Brexit — not blackmail.”

As British lawmakers acknowledged that they may have ask Brussels to allow a delay, European leaders warned that an extension would not be automatic.

French President Emmanuel Macron, speaking in a joint news conference with German Chancellor Angela Merkel, said a Brexit delay would need to be “justified” by “new British choices,” suggesting that May’s government would need to signal where this is all headed — versus asking for more time just because they are stuck.