diff --git a/cmux.go b/cmux.go
index 4595878..89cc910 100644
--- a/cmux.go
+++ b/cmux.go
@@ -197,19 +197,11 @@ func newMuxConn(c net.Conn) *MuxConn {
 // a non-zero number of bytes at the end of the input stream may
 // return either err == EOF or err == nil.  The next Read should
 // return 0, EOF.
-//
-// This function implements the latter behaviour, returning the
-// (non-nil) error from the same call.
 func (m *MuxConn) Read(p []byte) (int, error) {
-	n1, err := m.buf.Read(p)
-	if err == nil && m.buf.Len() == 0 {
-		err = io.EOF
+	if n, err := m.buf.Read(p); err != io.EOF {
+		return n, err
 	}
-	if n1 == len(p) || err != io.EOF {
-		return n1, err
-	}
-	n2, err := m.Conn.Read(p[n1:])
-	return n1 + n2, err
+	return m.Conn.Read(p)
 }
 
 func (m *MuxConn) getSniffer() io.Reader {
diff --git a/cmux_test.go b/cmux_test.go
index 669b713..90aaab9 100644
--- a/cmux_test.go
+++ b/cmux_test.go
@@ -278,12 +278,20 @@ func TestHTTP2(t *testing.T) {
 	if err != nil {
 		t.Fatal(err)
 	}
-	var b [len(http2.ClientPreface)]byte
-	if _, err := muxedConn.Read(b[:]); err != io.EOF {
-		t.Fatal(err)
+	{
+		var b [len(http2.ClientPreface)]byte
+		if _, err := muxedConn.Read(b[:]); err != nil {
+			t.Fatal(err)
+		}
+		if string(b[:]) != http2.ClientPreface {
+			t.Errorf("got unexpected read %s, expected %s", b, http2.ClientPreface)
+		}
 	}
-	if string(b[:]) != http2.ClientPreface {
-		t.Errorf("got unexpected read %s, expected %s", b, http2.ClientPreface)
+	{
+		var b [1]byte
+		if _, err := muxedConn.Read(b[:]); err != io.EOF {
+			t.Errorf("unexpected error %v, expected %v", err, io.EOF)
+		}
 	}
 }