user78: These physical coins are supposed to be worth 1 BTC each - they have a hologram and a private key that somehow gives access to a real (digital) Bitcoin.
What is the process of claiming this real (digital) BTC for the physical one?

user153: http://StrongCoin.com provides the ability to create an account from an imported MiniKey.  They have a fee of 1% of each spend, plus (currently) a 0.01 BTC network transaction fee.

user105: It is now possible to redeem a Casascius private key at MtGox if you have an account there.
Under "Add funds", choose "Redeem private key", then "SHA-256 Private Key". The full amount of the private key will be added to your MtGox account.

user5: I just want to add a few things about the Mini private key format.
This is essentially a compressed way to store a bitcoin private key. Casascius coins small, so the private key is represented by 22 characters printed like this:
S4b3N
3oGqDq
R5jNux
EvDwf

To get the real private key, you just have to calculate the SHA-256 hash of the string. There are many online tools to calculate hashes and most linux distributions can quickly do it from the terminal:
$ echo -n "S4b3N3oGqDqR5jNuxEvDwf" | sha256sum 
0c28fca386c7a227600b2fe50b7cae11ec86d3bf1fbe471be89827e19d72aa1d


user22: The first 8 characters of the public address are on each coin, which allows you to obtain the full address or check the balance at FirstBits or a similar service. Beneath the tamper-evident holographic sticker on each coin is the private address which corresponds to the public address. Once you've obtained both the public and private address, the process for importing this information into an existing wallet has already been documented in this prior question.
Edit: On a side note, it's probably worth noting that this process is almost identical to the process used to redeem BitBills
Edit 2: Apparently these coins use a different private key format than standard. Casascius has published a utility to convert these private keys to standard addresses and more about the "mini private key" format can be found at the wiki. For those lacking a copy of Visual Studio, I've compiled a copy for your convenience. These keys may also be computed using any calculator or code capable of generating an SHA256 hash:
SHA256("S4b3N3oGqDqR5jNuxEvDwf") = 0C28FCA386C7A227600B2FE50B7CAE11EC86D3BF1FBE471BE89827E19D72AA1D
Of course that's still in hex and needs to be converted to Bitcoin's modified Base58 but many wallet import tools will accept hex as well.
Edit 3: Just heard from Casascius that we may be able to redeem these coins, as well as any device bearing a mini private key directly at MtGox in the near future.
Edit 4: Support has officially been added

78: <p>These <a href="https://bitcointalk.org/index.php?topic=41892.0;all">physical coins</a> are supposed to be worth 1 BTC each - they have a hologram and a private key that somehow gives access to a real (digital) Bitcoin.</p>

<p>What is the process of claiming this real (digital) BTC for the physical one?</p>
 153: <p><a href="http://StrongCoin.com" rel="nofollow">http://StrongCoin.com</a> provides the ability to create an account from an imported MiniKey.  They have a fee of 1% of each spend, plus (currently) a 0.01 BTC network transaction fee.</p>
 105: <p>It is now possible to redeem a Casascius private key at <a href="http://www.mtgox.com" rel="nofollow">MtGox</a> if you have an account there.</p>

<p>Under "Add funds", choose "Redeem private key", then "SHA-256 Private Key". The full amount of the private key will be added to your MtGox account.</p>
 5: <p>I just want to add a few things about the Mini private key format.</p>

<p>This is essentially a compressed way to store a bitcoin private key. Casascius coins small, so the private key is represented by 22 characters printed like this:</p>

<pre><code>S4b3N
3oGqDq
R5jNux
EvDwf
</code></pre>

<p>To get the real private key, you just have to calculate the SHA-256 hash of the string. There are many <a href="http://www.xorbin.com/tools/sha256-hash-calculator">online tools</a> to calculate hashes and most linux distributions can quickly do it from the terminal:</p>

<pre><code>$ echo -n "S4b3N3oGqDqR5jNuxEvDwf" | sha256sum 
0c28fca386c7a227600b2fe50b7cae11ec86d3bf1fbe471be89827e19d72aa1d
</code></pre>
 22: <p>The first 8 characters of the public address are on each coin, which allows you to obtain the full address or check the balance at <a href="http://www.firstbits.com" rel="nofollow noreferrer">FirstBits</a> or a similar service. Beneath the tamper-evident holographic sticker on each coin is the private address which corresponds to the public address. Once you've obtained both the public and private address, the process for importing this information into an existing wallet has already been documented in <a href="https://bitcoin.stackexchange.com/questions/447/is-there-some-way-to-merge-two-wallets-or-to-import-export-addresses">this prior question</a>.</p>

<p><strong>Edit:</strong> <em>On a side note, it's probably worth noting that this process is almost identical to the process used to redeem BitBills</em></p>

<p><strong>Edit 2:</strong> <em>Apparently these coins use a different private key format than standard. Casascius has published a <a href="https://github.com/casascius/Bitcoin-Address-Utility" rel="nofollow noreferrer">utility</a> to convert these private keys to standard addresses and more about the "mini private key" format can be found at the <a href="https://en.bitcoin.it/wiki/Mini_private_key_format" rel="nofollow noreferrer">wiki</a>. For those lacking a copy of Visual Studio, I've <a href="http://dl.dropbox.com/u/20411213/BtcAddress.zip" rel="nofollow noreferrer">compiled a copy</a> for your convenience. These keys may also be computed using any calculator or code capable of generating an SHA256 hash:</em></p>

<p><code>SHA256("S4b3N3oGqDqR5jNuxEvDwf") = 0C28FCA386C7A227600B2FE50B7CAE11EC86D3BF1FBE471BE89827E19D72AA1D</code></p>

<p><em>Of course that's still in hex and needs to be converted to Bitcoin's modified Base58 but many wallet import tools will accept hex as well.</em></p>

<p><strong>Edit 3:</strong> <em>Just heard from Casascius that we may be able to redeem these coins, as well as any device bearing a mini private key directly at MtGox in the near future.</em></p>

<p><strong>Edit 4:</strong> <em>Support has officially been <a href="https://bitcointalk.org/index.php?topic=46908.0;all" rel="nofollow noreferrer">added</a></em></p>
