If I understand it right, I won't be able to use it till the BitCoin Core finish synchronising.
Is there anything I can do to use the money now? From the sender site transaction was completed succesfully: https://www.blocktrail.com/BTC/address/1CQcKeg9L4zAkNhVaKMntS9kTY8n1rpcqA
And I'm always out of disk space because of sychnhronisng.
1CQcKeg9L4zAkNhVaKMntS9kTY8n1rpcqA
You've just stepped into one of the worst problems with Bitcoin. The "default wallet", Bitcoin Core, is not for beginners.
You'll probably want to use a different wallet instead. The problem is that you've already sent a payment to your Bitcoin Core Wallet.
To recover your funds, you'll need to take a crash course in Bitcoin internals.
Your Core wallet contains private keys. You can move private keys from one wallet to another, although this is really bad practice and should only be used when nothing else will work. This might be one of those times, depending on how much money is at stake and how unwilling you might be to let Bitcoin Core continue syncing for the next several days, but as you mentioned you're running out space so it will never complete.
You sent funds to an address (a long sequence of text beginning with "1").
To get the private key for your address, get to the Core console (Help->Debug window).
Next, click on the "Console" tab. There is a text box at the bottom with a ">" symbol and a blinking cursor.
Enter this into the box:
dumpprivkey <address>
where <address> is your address.
From there, you have the private key (begins with "K" or "L", possibly "5"), which you can import into another wallet.
Elsewhere, you indicated you're using electrum. Here's how to "sweep" funds from your private key into Electrum:
http://bitzuma.com/posts/a-beginners-guide-to-the-electrum-bitcoin-wallet/#sweep-a-paper-wallet
Note that the article talks about a paper wallet, which is simply a private key printed on paper.
