The announcement comes just one day after French media reports that Tour de France organisers were seeking to block Froome from riding in this year's race.
Froome had been racing under the cloud of a potential ban after a urine sample he provided at the Spanish Vuelta in September 2017 showed a concentration of the asthma drug salbutamol that was twice the permitted level.
"I am very pleased that the UCI has exonerated me", Froome said after the Union Cycliste Internationale, the sport's ruling body, said a probe into Froome had been dropped.
Froome exercised his right to prove his abnormal result was the effect of a permitted use of the asthma medication and submitted his explanation to the UCI with additional expert evidence after requesting information from the World Anti-Doping Agency (WADA).
"In light of WADA's unparalleled access to information and authorship of the salbutamol regime, the UCI has decided, based on WADA's position, to close the proceedings against Froome".
"We were awaiting this decision [from the UCI] and now he has been cleared", said Prudhomme.
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In its own statement WADA confirmed that it would no longer pursue Froome's adverse analytical finding for salbutamol (AAF) at the 2017 Vuelta a Espana, going as far as saying that the original test results did not comprise an AAF.
Froome's bid to join Hinault on a record-equalling five yellow jerseys begins in Noirmoutier on Saturday. "So be it. He will be on the starting line because the UCI and WADA finally gave an answer".
Froome said he had "always known" he had not broken any rules because he "never took more than the maximum permissible amount" of 1600 micrograms over 24 hours, or 800 mcg over 12 hours - roughly equating to 16 or eight puffs of an inhaler.
Monday's decision also means Froome will be able to hold onto the Giro d'Italia trophy he won in May, which gave him three straight Grand Tour titles.
But Brailsford said they had made the correct decision based on "total confidence in Chris and his integrity".
"A review of all Chris's 21 test results from the Vuelta revealed that the stage 18 result was within his expected range of variation and therefore consistent with him having taken a permitted dose of salbutamol".
Since those cases, a 2016 scientific study showed that some subjects who inhaled the maximum dose all at once exceeded the "decision limit" of 1200ng/mL salbutamol levels in urine that would trigger an adverse analytical finding, but if the samples were standardized for concentration (specific gravity), there were fewer samples that exceeded the limit.