Robinson, 33, who was released by relegated Bolton Wanderers at the end of last season, has signed a one-month deal that runs until 26 October.
Clark said: "I'd like to thank the board for allowing me to bring in Paul.
"He is an experienced defender who will enhance our playing squad."
Already without injured skipper Stephen Carr, who has a career-threatening knee injury, Clark last week lost season-long loan signing Paul Caddis, who is expected to be out until Christmas at the earliest once he has had shoulder surgery.
The injuries have left us with just two fit senior defenders
And, after losing David Murphy with a torn hamstring midway through the first half of Saturday's 5-0 home defeat by Barnsley, the Blues boss is now also without Pablo Ibanez, who is expected to be out for four weeks after breaking a bone in his foot.
"While I have been impressed with the progress of our young defenders, the injuries have left us with just two fit senior defenders," added Clark. "So the acquisition of Paul was one we needed to make.
"As a senior professional, Paul will give us that added experience which is always useful in this competitive division."
Former England Under-21 international Robinson began his career with home-town club Watford before being signed by Gary Megson for Albion in 2003.
After six years at The Hawthorns, a season-long loan move to Bolton was made permanent in 2010.
But Robinson, who is expected to make his Blues debut in Saturday evening's clash with leaders Brighton at the Amex Stadium, has not played since finishing last season out on loan with Leeds United.

The South Africa-born 21-year-old right-hander will now remain at Lord's until the end of the 2017 season.
Eskinazi was a regular for Middlesex's Second XI this year and made his first-class debut in last month's victory over Yorkshire at Lord's.
"Stephen thoroughly deserves a new contract," managing director of cricket Angus Fraser said.
"Once again he was our top run-scorer in the second eleven and was rewarded for this with a first-class debut.
"He looked comfortable against high quality bowling, which is a good sign for the future."

Hemmings scored 24 goals for Cowdenbeath last season, including two hat-tricks, and was named Scottish Championship Player of the Season.
Barnsley manager Danny Wilson said he did not believe the 22-year-old would struggle to adapt to English football.
"He has got good pace and is a very good finisher as the stats will tell you," said Wilson.
"He can play in a number of positions, but more importantly he's down the middle and has got goals in him, which is what we want to bring into the team."
Hemmings began his career at Tamworth before moving to Rangers, for whom he made 10 appearances, and then joined Cowdenbeath in the summer of 2013 after a successful loan spell.
He is Barnsley's second signing since being relegated from the Championship, following the arrival of ex-Sheffield Wednesday goalkeeper Adam Davies.

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Goals from Gareth Furlong, Luke Hawker and Rupert Shipperley secured the win in Belfast.
They play Scotland on Sunday before facing France in their final group game on Tuesday,
A top-two finish in the eight-team tournament would lift them in the rankings and give them a chance of a first World Cup appearance.
The World League is a series of tournaments held over a two-year period with the latest edition running from Saturday 11 to Sunday 19 March in Belfast.
The final group standings decide the knockout fixtures and the two teams who reach the final will go through to the next round of the World League in London in June.
Birmingham City boss Lee Clark has responded to his side's string of defensive injuries by bringing in former West Bromwich Albion left-back Paul Robinson on a short-term deal.
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Middlesex wicketkeeper-batsman Stephen Eskinazi has signed a new two-year contract with the club.
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Relegated Barnsley have completed the signing of striker Kane Hemmings from Cowdenbeath on a free transfer.
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Wales men's hockey team beat Poland in their opening game in the of the World League 2 tournament.