The Office for National Statistics said industrial output fell 0.5% compared with February, when it dropped by a revised 0.8%.
Falling demand for energy as a result of warmer weather was behind the change, the ONS said.
Construction output fell by 0.7% in March, after a 1.7% fall in February.
Meanwhile, the UK's deficit in goods and services widened to Â£4.9bn in March, from a revised figure of Â£2.6bn in February.
The ONS said rising oil imports, as well as imports of machinery and cars, had contributed to the unexpectedly high figure.
"March's simply dreadful trade figures demonstrate that Britain is failing to capitalise on sterling's depreciation," said Samuel Tombs, chief UK economist at Pantheon Macroeconomics.
"The trade deficit is extremely volatile, but has shown no underlying improvement since sterling began to depreciate in late 2015."
Howard Archer, chief UK and European economist at IHS Markit, described the figures as "a ropey set of March data for the UK economy that point to a poor end to a disappointing first quarter".
He added: "The poor data dilute any hopes that markedly slower GDP growth of 0.3% quarter-on-quarter in the first quarter could be revised up.
"Indeed, the actual industrial production data was weaker than estimated in the preliminary first-quarter GDP estimate, while the trade deficit unexpectedly widened."
Activity in the UK's industrial and construction sectors shrank in March, new figures show.