Python 3

import base64
from io import BytesIO

buffered = BytesIO()
image.save(buffered, format="JPEG")
img_str = base64.b64encode(buffered.getvalue())

Python 2

import base64
import cStringIO

buffer = cStringIO.StringIO()
image.save(buffer, format="JPEG")
img_str = base64.b64encode(buffer.getvalue())

As suggested in the comments, I tried pyvips as below:

#!/usr/bin/env python3
import requests
import base64
import numpy as np
from PIL import Image
from io import BytesIO
from cv2 import imencode
import pyvips

def vips_2PNG(image,compression=6):
    # Convert PIL Image to Numpy array
    na = np.array(image)
    height, width, bands = na.shape

    # Convert Numpy array to Vips image
    dtype_to_format = {
       'uint8': 'uchar',
       'int8': 'char',
       'uint16': 'ushort',
       'int16': 'short',
       'uint32': 'uint',
       'int32': 'int',
       'float32': 'float',
       'float64': 'double',
       'complex64': 'complex',
       'complex128': 'dpcomplex',
    }
    linear = na.reshape(width * height * bands)
    vi = pyvips.Image.new_from_memory(linear.data, width, height, bands,dtype_to_format[str(na.dtype)])

    # Save to memory buffer as PNG
    data = vi.write_to_buffer(f".png[compression={compression}]")
    return data

def vips_including_reading_from_disk(image):
    # Load image from disk
    image = pyvips.Image.new_from_file('stuttgart.png', access='sequential')
    # Save to memory buffer as PNG
    data = image.write_to_buffer('.png')
    return data

def faster(image):
    image_arr = np.array(image)
    _, byte_data = imencode('.png', image_arr)        
    return byte_data

def orig(image, faster=True):    
    output_buffer = BytesIO()
    image.save(output_buffer, format='PNG')
    byte_data = output_buffer.getvalue()
    return byte_data

# img_url = "https://www.cityscapes-dataset.com/wordpress/wp-content/uploads/2015/07/stuttgart03.png"
filename = 'stuttgart.png'
img = Image.open(filename)

# r = orig(img)
# print(len(r))
# %timeit r = orig(img)

# r = faster(img)
# print(len(r))
# %timeit r = faster(img)

# r = vips_including_reading_from_disk(filename)
# print(len(r))
# %timeit r = vips_including_reading_from_disk(filename)

# r = vips_2PNG(img,0)
# print(len(r))
# %timeit r = vips_2PNG(img,0)

I was looking at trading off the compression parameter between file size and speed. Here is what I got - I wouldn't compare absolute values, but rather look at the performance relative to each other on my machine:

               Filesize        Time
PIL            1.7MB           1.12s
OpenCV         2.0MB           173ms   <--- COMPARE
vips(comp=0)   6.2MB           66ms
vips(comp=1)   2.0MB           132ms   <--- COMPARE
vips(comp=2)   2.0MB           153ms

I have put arrows next to the ones I would compare.

Here's a short but complete demo of your code using a ByteIO instead of StringIO. I've also added a function to do the reverse conversion. It runs correctly on Python 2.6 and 3.6. The only difference is that in Python 3 the Base64 output is a b string.

from PIL import Image
from io import BytesIO
import base64

# Convert Image to Base64 
def im_2_b64(image):
    buff = BytesIO()
    image.save(buff, format="JPEG")
    img_str = base64.b64encode(buff.getvalue())
    return img_str

# Convert Base64 to Image
def b64_2_img(data):
    buff = BytesIO(base64.b64decode(data))
    return Image.open(buff)

# Test

img = Image.new('RGB', (120, 90), 'red')
img.show()

img_b64 = im_2_b64(img)
print(img_b64)

new_img = b64_2_img(img_b64)
new_img.show()

Python 3 output

b'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'

I found the solution. Hope this helps !

img = Image.fromarray(data, 'RGB')                  #Crée une image à partir de la matrice
buffer = BytesIO()
img.save(buffer,format="JPEG")                  #Enregistre l'image dans le buffer
myimage = buffer.getvalue()                     
print "data:image/jpeg;base64,"+base64.b64encode(myimage)

You need to base64 encode before you can decode.

You can achieve this without creating a temporary file by using an in memory file, with io.BytesIO()

in_mem_file = io.BytesIO()
img.save(in_mem_file, format = "PNG")

img_bytes = in_mem_file.getvalue()

base64_encoded_result_bytes = base64.b64encode(img_bytes)
# b64encode returns a byte string so needs converting back to a string
base64_encoded_result_str = base64_encoded_result_bytes.decode('ascii')

ImageGrab.grabclipboard() returns an Image object. You need to convert it to a known image format, like jpeg or png, then to encode the resulting string in base64 to be able to use it within an HTML img tag:

import cStringIO

jpeg_image_buffer = cStringIO.StringIO()
image.save(jpeg_image_buffer, format="JPEG")
imgStr = base64.b64encode(jpeg_image_buffer.getvalue())

(answer has been edited to fix the typo).