Biggar notched another penalty in the second half but replacement Geoffrey Palis took over kicking duties for the French side in the dying moments to bag another three points and a Castres victory.
It leaves Ospreys rock bottom of Pool 1 with one point and almost no hope of qualifying, but Rees expects his players to rally and produce another brave effort at the Liberty Stadium on Friday.
"There were opportunities at the end to win but it didn't happen and we have to look forward to the next game," said Rees.
"Effectively we are out of the competition. But it was a huge effort by all the boys, from Alun Wyn Jones to young Daniel Suter - they all gave it everything.
"We had a very difficult week with only one full training day but the boys applied themselves and gave everything they could for the shirt."
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