As a multilayer ceramic electronic element, which is of interest to the present invention, for example, a multilayer thermistor with a positive temperature coefficient may be mentioned.  The multilayer thermistor with a positive temperature coefficient has the following structure. 
First, the multilayer thermistor with a positive temperature coefficient comprises a laminate used as an element body.  The laminate comprises a plurality of thermistor layers having a positive temperature coefficient laminated to each other, and a plurality of internal electrodes formed along specific interfaces between the thermistor layers.  The internal electrodes are disposed in the lamination direction so as to extend alternately to one end surface and the other end surface of the laminate. 
In addition, the multilayer thermistor with a positive temperature coefficient also comprises external electrodes functioning as terminals on the respective end surfaces of the laminate described above.  The external electrodes are electrically connected to the internal electrodes at the respective end surfaces of the laminate. 
The multilayer thermistor with a positive temperature coefficient described above is manufactured, for example, by a manufacturing process shown in FIG.  3, as disclosed in Japanese Unexamined Patent Application Publication No.  5-308003. 
As shown in FIG.  3, first, step 1, forming a green laminate, is performed.  The green laminate obtained in this step is to be formed into the sintered laminate described above by firing and comprises thermistor green layers to be formed into the thermistor layers and conductive paste layers to be formed into the internal electrodes. 
In general, the green laminates are formed by the steps of forming thermistor green sheets which are to be formed into thermistor green layers, cutting the thermistor green sheets so as to have predetermined dimensions, printing a conductive paste on the thermistor green sheets in order to form conductive paste layers which are to be formed into the internal electrodes, then laminating the thermistor green sheets to each other, followed by pressing to form a green mother laminate, and cutting this green mother laminate so as to form green laminates having predetermined dimensions. 
The conductive paste layers for the internal electrodes described above are formed by using a conductive paste containing nickel as a conductive component, which is an inexpensive base metal and which can have an ohmic contact with the thermistor layer. 
Next, step 2, firing the green laminate, is performed.  When a base metal such as nickel is used as the conductive component of the internal electrode as described above, this firing step 2 is performed in a reducing atmosphere in order to prevent the base metal from being oxidized.  Hence, in this case, after firing step 2, heat treatment (reoxidation) is performed in an oxidizing atmosphere so that the thermistor layers are able to have positive temperature coefficient properties.  By this firing step 2, the sintered laminate can be obtained. 
Next, wet-barrel step 3 is performed.  This wet-barrel step 3 is generally performed in a manufacturing process not only for a multilayer thermistor with a positive temperature coefficient but also for a chip-type ceramic electronic element.  In this step, the ceramic element bodies after firing (that is, the sintered bodies) are mixed and stirred with a polishing medium such as powdered alumina and water for barrel polishing (wet barrel) in order to prevent cracking, so-called chipping, of ceramic element bodies.  As a result, the corners and ridgelines of the sintered ceramic element bodies, in other words, the laminates, can be rounded. 
Next, step 4, applying an external electrode paste, is performed.  That is, a conductive paste for forming the external electrodes is applied onto the respective end surfaces of the sintered laminate, and conductive paste films are formed thereby.  In this step, a conductive component of the external electrode preferably contains the same metal as that of the conductive component of the internal electrode in order to obtain good electrical conduction state with the internal electrode.  Hence, as described above, when the internal electrode contains nickel, a material containing nickel is preferably used for the conductive paste for this external electrode. 
Next, step 5, firing the external electrodes, is performed.  In this step, this firing step 5 is performed in a reducing atmosphere, when the conductive paste film for the external electrode contains a base metal such as nickel. 
Through the steps described above, the multilayer thermistor with a positive temperature coefficient is obtained. 
However, in the manufacturing process shown in FIG.  3, the following problems may arise in some cases. 
Step 4 of applying the external electrode paste is performed after firing step 2.  In addition, the internal electrodes in the sintered laminate obtained through firing step 2 may withdraw by contraction from the end surfaces of the laminate to the inside thereof in some cases, and as a result, they may not extend to the end surfaces.  Hence, in step 4 of applying the external electrode paste, the conductive paste films may not be appropriately connected to the internal electrodes when being formed for the external electrodes in some cases. 
In addition, when a base metal such as nickel is used as the conductive component of the internal electrode and as the conductive component of the external electrode, as described above, firing step 2 must be performed in a reducing atmosphere, and in addition, step 5 of firing the external electrodes must also be performed in a reducing atmosphere.  Compared to the case in which an oxidizing atmosphere is obtained, the cost therefor is very high when a reducing atmosphere is obtained.  Hence, when a reducing atmosphere is necessary in both steps 2 and 5, the cost of mass production is increased. 
As a method capable of solving the problem described above, a manufacturing method shown in FIG.  4 may be mentioned. 
As shown in FIG.  4, first, step 11 of manufacturing a green laminate is performed.  This step 11 of manufacturing the green laminate is performed in a manner substantially equivalent to that of step 1 of manufacturing the green laminate shown in FIG.  3. 
Next, step 12 of applying an external electrode paste is performed.  This step 12 of applying the external electrode paste is substantially equivalent to step 4 of applying the external electrode paste shown in FIG.  3 except that the step is performed on the green laminate.  However, since the conductive paste layers for the internal electrodes provided in the green laminate are not yet contracted by firing, appropriate connection states between the internal electrodes and the external electrodes can be achieved. 
Next, firing step 13 is performed.  In this firing step, the green laminate is fired together with the conductive paste films for the external electrodes.  When the conductive paste layers for the internal electrodes and the conductive paste films for the external electrodes contain a base metal such as nickel, firing step 13 is performed in a reducing atmosphere, and subsequently, the fired laminate is heat-treated in an oxidizing atmosphere.  As described above, since the conductive paste films for the external electrodes and the green laminate are simultaneously fired in firing step 13, the control for obtaining the reducing atmosphere is only necessary in this firing step 13; hence, compared to the manufacturing method shown in FIG.  3, the cost can be reduced. 
Next, wet-barrel step 14 is performed.  This wet-barrel step 14 is performed in a manner substantially equivalent to that of wet-barrel step 3 shown in FIG.  3, and by wet-barrel polishing, the corners and ridgelines of the sintered laminate are rounded for chipping prevention. 
However, there are still problems which have to be solved in the manufacturing method shown in FIG.  4. 
That is, since wet-barrel step 14 is performed for the sintered laminate provided with the external electrodes, parts of the external electrodes are polished by barrel polishing, and as a result, the conductions between the external electrodes and the internal electrodes may become unstable in some cases. 
In addition, since wet-barrel steps 3 and 14 are performed after firing steps 2 and 13 in both manufacturing methods shown in FIGS.  3 and 4, the barrel polishing is performed on the sintered laminate.  Hence, a problem may arise in that cracking or the like is liable to occur in the sintered laminate, because of the barrel polishing. 
In addition to the case in which the multilayer thermistor with a positive temperature coefficient described above is manufactured, the same problem as described above may also arise when other multilayer ceramic electronic elements are manufactured each having a structure similar to that of the multilayer thermistor with a positive temperature coefficient.