ECB mode will produce the same result for identical blocks (ie: 16 bytes for AES). An attacker could be able to guess the encrypted message. The use of AES in CBC mode with a HMAC is recommended guaranteeing integrity and confidentiality.

Risk

The ECB mode will produce identical encrypted block for equivalent plain text block. This could allow an attacker that is eavesdropping to guess the content sent. This same property can also allow the recovery of the original message. Futhermore, this cipher mode alone does not guarantee integrity.

Vulnerable Code

using (var aes = new AesManaged {
    KeySize = KeyBitSize,
    BlockSize = BlockBitSize,
    Mode = CipherMode.ECB,
    Padding = PaddingMode.PKCS7
})
{
    using (var encrypter = aes.CreateEncryptor(cryptKey, new byte[16]))
    using (var cipherStream = new MemoryStream())
    {
        using (var cryptoStream = new CryptoStream(cipherStream, encrypter, CryptoStreamMode.Write))
        using (var binaryWriter = new BinaryWriter(cryptoStream))
        {
            //Encrypt Data
            binaryWriter.Write(secretMessage);
        }
        cipherText = cipherStream.ToArray();
    }
}

Solution

Using bouncy castle:

 
using Org.BouncyCastle.Crypto;
using Org.BouncyCastle.Crypto.Engines;
using Org.BouncyCastle.Crypto.Generators;
using Org.BouncyCastle.Crypto.Modes;
using Org.BouncyCastle.Crypto.Parameters;
using Org.BouncyCastle.Security;

public static readonly int BlockBitSize = 128;
public static readonly int KeyBitSize = 256;

public static byte[] SimpleEncrypt(byte[] secretMessage, byte[] key)
{
    //User Error Checks
    if (key == null || key.Length != KeyBitSize / 8)
        throw new ArgumentException(String.Format("Key needs to be {0} bit!", KeyBitSize), "key");

    if (secretMessage == null || secretMessage.Length == 0)
        throw new ArgumentException("Secret Message Required!", "secretMessage");

    //Using random nonce large enough not to repeat
    var nonce = new byte[NonceBitSize / 8];
    Random.NextBytes(nonce, 0, nonce.Length);

    var cipher = new GcmBlockCipher(new AesFastEngine());
    var parameters = new AeadParameters(new KeyParameter(key), MacBitSize, nonce, new byte[0]);
    cipher.Init(true, parameters);

    //Generate Cipher Text With Auth Tag
    var cipherText = new byte[cipher.GetOutputSize(secretMessage.Length)];
    var len = cipher.ProcessBytes(secretMessage, 0, secretMessage.Length, cipherText, 0);
    cipher.DoFinal(cipherText, len);

    //Assemble Message
    using (var combinedStream = new MemoryStream())
    {
        using (var binaryWriter = new BinaryWriter(combinedStream))
        {
            //Prepend Nonce
            binaryWriter.Write(nonce);
            //Write Cipher Text
            binaryWriter.Write(cipherText);
        }
        return combinedStream.ToArray();
    }
}

Custom implementation of Encrypt and HMAC

    
using System.IO;
using System.Security.Cryptography;
public static byte[] SimpleEncrypt(byte[] secretMessage, byte[] cryptKey, byte[] authKey, byte[] nonSecretPayload = null)
{
    //User Error Checks
    if (cryptKey == null || cryptKey.Length != KeyBitSize / 8)
        throw new ArgumentException(String.Format("Key needs to be {0} bit!", KeyBitSize), "cryptKey");

    if (authKey == null || authKey.Length != KeyBitSize / 8)
        throw new ArgumentException(String.Format("Key needs to be {0} bit!", KeyBitSize), "authKey");

    if (secretMessage == null || secretMessage.Length < 1)
        throw new ArgumentException("Secret Message Required!", "secretMessage");

    byte[] cipherText;
    byte[] iv;
    using (var aes = new AesManaged {
        KeySize = KeyBitSize,
        BlockSize = BlockBitSize,
        Mode = CipherMode.CBC,
        Padding = PaddingMode.PKCS7
    })
    {
        //Use random IV
        aes.GenerateIV();
        iv = aes.IV;
        using (var encrypter = aes.CreateEncryptor(cryptKey, iv))
        using (var cipherStream = new MemoryStream())
        {
            using (var cryptoStream = new CryptoStream(cipherStream, encrypter, CryptoStreamMode.Write))
            using (var binaryWriter = new BinaryWriter(cryptoStream))
            {
            //Encrypt Data
            binaryWriter.Write(secretMessage);
            }
            cipherText = cipherStream.ToArray();
        }
    }
    //Assemble encrypted message and add authentication
    using (var hmac = new HMACSHA256(authKey))
    using (var encryptedStream = new MemoryStream())
    {
        using (var binaryWriter = new BinaryWriter(encryptedStream))
        {
            //Prepend IV
            binaryWriter.Write(iv);
            //Write Ciphertext
            binaryWriter.Write(cipherText);
            binaryWriter.Flush();
            //Authenticate all data
            var tag = hmac.ComputeHash(encryptedStream.ToArray());
            //Postpend tag
            binaryWriter.Write(tag);
        }
        return encryptedStream.ToArray();
    }
}

References

Padding Oracles for the masses (by Matias Soler)
Wikipedia: Authenticated encryption
NIST: Authenticated Encryption Modes
CAPEC: Padding Oracle Crypto Attack
CWE-696: Incorrect Behavior Order