Unknown. As far as I can tell this has never been confirmed in any other sources, and Kreacher's behaviour afterwards is that of a happy-to-serve elf, free or no.
There's some quite weak evidence in the following chapter that he's still magically bound, when Harry gives him an order and he instantly complies, to what is obviously his detriment (emphasis mine):
There was the sound of pattering feet, a blaze of shining copper, an echoing clang, and a shriek of agony: Kreacher had taken a run at Mundungus and hit him over the head with a saucepan.
"Call 'im off, call 'im off, 'e should be locked up!" screamed Mundungus, cowering as Kreacher raised the heavy-bottomed pan again.
"Kreacher, no!" shouted Harry.
Kreacher's thin arms trembled with the weight of the pan, still held aloft.
Harry Potter and the Deathly Hallows Chapter 11: "The Bribe"
But we can't be completely certain.
Slightly stronger evidence is that this possibility doesn't seem to occur to anybody. Though the reasons for this oversight aren't stated explicitly, we can easily imagine some:
Jewelry doesn't count as clothes
This seems a sensible assumption, but to my knowledge is not made explicit.
Harry is hoping that loyalty will trump binding magical contracts
It's clear that, magical contract or no, Kreacher will help the people he's loyal to, to whatever extent he's permitted to do so; this is something Hermione points out explicitly:
What do wizard wars mean to an elf like Kreacher? He's loyal to people who are kind to him, and Mrs. Black must have been, and Regulus certainly was, so he
served them willingly and parroted their beliefs.
I'm sure 'Miss Cissy' and 'Miss Bella' were perfectly lovely to Kreacher when he turned up, so he did them a favor and told them everything they wanted to know.
Harry Potter and the Deathly Hallows Chapter 10: "Kreacher's Tale"
Treating Kreacher with kindness will induce him to help them, freedom or no. That's the hope, anyway.
It's actually a fairly low-risk plan
This isn't a reason, per se, but consider the costs versus the benefits of giving Kreacher the locket. Having Kreacher bound to Harry's service isn't actually that helpful at the moment, and it's already demonstrably harmful. Setting him free is going to have one of two outcomes:
He's disloyal, and runs off to Bellatrix. But, so what? The reason Dumbledore talks Harry out of freeing him in Half-Blood Prince is because of all the sensitive information the elf has been presumably overhearing, but what does any of that matter anymore? The Order has abandoned Grimmauld Place, so he doesn't have any fresh intelligence, and anything he did know is fairly stale.
This is, of course, presuming that he can't tell the Death Eaters where Grimmauld Place is, but why should they care about that anyway? As far as they know, Snape has already spilled the beans1.
He's loyal, and he stays despite being freed. See the previous section for why this seems like a likely outcome.
1 Probably, maybe