diff --git a/Lib/tarfile.py b/Lib/tarfile.py
--- a/Lib/tarfile.py
+++ b/Lib/tarfile.py
@@ -689,7 +689,10 @@
             length = min(size, stop - self.position)
             if data:
                 self.fileobj.seek(offset + (self.position - start))
-                buf += self.fileobj.read(length)
+                b = self.fileobj.read(length)
+                if len(b) != length:
+                    raise ReadError("unexpected end of archive")
+                buf += b
             else:
                 buf += NUL * length
             size -= length
@@ -2243,8 +2246,20 @@
             self.firstmember = None
             return m
 
+        # Advance the file position if needed.
+        size = self.offset - self.fileobj.tell()
+
+        if size < 0:
+            self.fileobj.seek(0)
+            size = self.offset
+
+        while size > 0:
+            s = min(size, RECORDSIZE)
+            if len(self.fileobj.read(s)) != s:
+                raise ReadError("unexpected end of archive")
+            size -= s
+
         # Read the next block.
-        self.fileobj.seek(self.offset)
         tarinfo = None
         while True:
             try:
diff --git a/Lib/test/test_tarfile.py b/Lib/test/test_tarfile.py
--- a/Lib/test/test_tarfile.py
+++ b/Lib/test/test_tarfile.py
@@ -349,6 +349,24 @@
             finally:
                 tar.close()
 
+    def test_premature_end_of_archive(self):
+        for size in (512, 600, 1024, 1200):
+            with tarfile.open(tmpname, "w:") as tar:
+                t = tarfile.TarInfo("foo")
+                t.size = 1024
+                tar.addfile(t, io.BytesIO(b"a" * 1024))
+
+            os.truncate(tmpname, size)
+
+            with tarfile.open(tmpname) as tar:
+                with self.assertRaises(tarfile.ReadError):
+                    for t in tar:
+                        pass
+
+            with tarfile.open(tmpname) as tar:
+                with self.assertRaises(tarfile.ReadError):
+                    tar.extractfile(tar.next()).read()
+
 
 class MiscReadTestBase(CommonReadTest):
     def requires_name_attribute(self):