Although it may seem hard to buy bitcoin without a bank account, services that allow the unbanked to buy bitcoin often deliver your coins faster. Bitcoin purchases made with cash often have delivery times of under 2 hours. Bitcoin purchases made with credit/debit card or ACH transfer often have wait times of up to 5 days before the coins are delivered.
Bitcoin can be a great tool for the unbanked, so it's no surprise that many companies are focused on helping those with no bank account buy bitcoin.
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Local Bitcoins provides escrow and offers a number of payment methods for buying bitcoin with no bank account. Cash deposit, Western Union, face-to-face cash trade, Amazon gift card are all offered on top of many other payment methods. Local Bitcoins provides escrow, but be sure to buy from sellers with previous trade history and positive feedback.
Wall of Coins is a peer-to-peer Bitcoin exchange that offers a number of payment methods. Cash deposit, however, is the exchange's most common payment method. There are over 100,000 deposit locations available across the United States. Wall of Coins also supports Canada, Germany, Poland, Argentina, Latvia, Poland, and the Philippines. After a cash deposit is made you'll usually receive your bitcoins within 15 minutes.
BitQuick provides escrow between buyers and sellers. Payments are made via cash deposit at bank locations across the U.S., and via SEPA transfer in the EU. Your coins will usually be delivered in under two hours if payment was made via cash deposit.
LibertyX lets you buy bitcoin with cash at a number of retail stores across the United States. You'll have to verify your identity before buying, making LibertyX less private than some of the other options. You can buy up to $1000 worth of bitcoin. You'll be charged a 1% fee on all purchases, but if you sign up with Facebook you can buy your first $1000 fee free. Your bitcoin should arrive a few minutes after your payment is made.
Coin ATM Radar helps you find Bitcoin ATMs in your area, where you can by bitcoins with cash and no bank account.
Mycelium is a Bitcoin wallet for iOS and Android. The Android version of the app has a "local trader" feature that helps connect you with locals who are selling Bitcoin. You can arrange to meet face-to-face and buy bitcoin with cash.
The availability of the services listed above may vary by location.