Webber, who will also take the role of accounting officer, was until recently the chief executive of the Criminal Injuries Compensation Authority. Prior to his seven years there, his positions include head of policy and planning at the Arts Council.
Webber joins on October 25, allowing acting chief executive Dr Ian Leigh to return to his normal role of managing director.
Millie Banerjee, chair of Postwatch, said: "I am delighted that we have been able to attract a candidate of Howard's calibre and look forward to working with him on the challenges ahead."
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