-----------------------
Class
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@RestResource(urlMapping='/leadconvert/*')
global with sharing class RestLeadConvert {
    public static final Integer StatusCodeOK = 200;
    public static final Integer StatusCodeBadRequest = 400;
    public static final Integer StatusCodeInternalServerError = 500;
    public static String DEFAULT_CONVERTED_STATUS = [SELECT Id, MasterLabel FROM LeadStatus WHERE IsConverted=true limit 1].MasterLabel;

    public class BadRequestException extends Exception {}

    // Represents a REST response
    class ResponseObject {
        String message;
        String code;
        Database.LeadConvertResult lead_convert_result;

        public Blob blob() {
            return Blob.valueOf(JSON.serialize(this));
        }

        public ResponseObject(String message, String code, Database.LeadConvertResult lcr) {
            this(message, code);
            this.lead_convert_result = lcr;
        }

        public ResponseObject(String message, String code) {
            this.message = message;
            this.code = code;
        }
    }

    @HttpPost
    global static void doPost() {
        RestContext.response.addHeader('Content-Type', 'application/json');

        Database.LeadConvertResult lcr ;
        try {
            Database.LeadConvert lc = parseRequest(RestContext.request);
            lc.setLeadId(parseTrailingLeadId(RestContext.request.requestURI));

            lcr = Database.convertLead(lc);
        }
        catch (Exception e) {
            RestContext.response.responseBody = new ResponseObject(e.getMessage(), exceptionTypeNameToCode(e)).blob();

            if (e.getTypeName() == 'RestLeadConvert.BadRequestException') {
                RestContext.response.statusCode = StatusCodeBadRequest;
            }
            else {
                // any other exception is unexpected; dump the exception and a stack trace to help debug it in production
                system.debug(e);
                system.debug(e.getStackTraceString());
                RestContext.response.statusCode = StatusCodeInternalServerError;
            }
            return;
        }
    
        if (lcr.isSuccess()) {
            RestContext.response.responseBody = new ResponseObject('Success', 'success', lcr).blob();
            RestContext.response.statusCode = StatusCodeOK;
        }
        else {
            RestContext.response.responseBody = new ResponseObject('Fail', 'see lead_convert_result', lcr).blob();
            RestContext.response.statusCode = StatusCodeInternalServerError;
        }
    }

    // Given an Exception typename like 'RestLeadConvert.BadRequestException', return 'BadRequestException'
    static String exceptionTypeNameToCode(Exception e) {
        String[] qualifiedName = e.getTypeName().split('\\.');
        return qualifiedName[qualifiedName.size()-1];
    }
    
    // completely parse the request body into an instance of LeadConvertBody
    // throw exception if it fails to parse in any way
    @TestVisible static Database.LeadConvert parseRequest(RestRequest request) {
        if (request.requestBody == null) {
            return createDefaultLeadConvert();
        }

        try {
            return (Database.LeadConvert)JSON.deserialize(request.requestBody.toString(), Database.LeadConvert.class);
        }
        catch(Exception e) {
            throw new BadRequestException('Failed to deserialize json body (%)'.replace('%', e.getMessage()));
        }
    }

    // this determines the behavior for a POST with no request body
    static Database.LeadConvert createDefaultLeadConvert() {
        Database.LeadConvert lc = new Database.LeadConvert(); 
        lc.setConvertedStatus(DEFAULT_CONVERTED_STATUS);
        return lc;
    }

    // get the LeadId from the requestURI and throw an exception if impossible
    static Id parseTrailingLeadId(String requestURI) {
        String leadId = getTrailingLeadId(requestURI);
        if (leadId == null) {
            throw new BadRequestException('requestURI does not have a valid trailing Lead Id');
        }
        return leadId;
    }

    // get the LeadId from the requestURI and return null if impossible
    @TestVisible
    static Id getTrailingLeadId(String requestURI) {
        Integer lastSlashIndex = requestURI.lastIndexOf('/');
        if (lastSlashIndex != -1) {
            String leadId = requestURI.substring(lastSlashIndex+1);
            if (leadId instanceOf Id && ((ID)leadId).getsobjecttype() == Schema.Lead.getSObjectType()) {
                return (Id)leadId;
            }
        }

        return null;
    }
}

-----------------------------------
Test Class
-----------------------------------
@isTest
public class RestLeadConvertTest{
    @isTest static void testDoPost()
    {
        Lead lead = new Lead (FirstName = 'John', LastName = 'Test', Company = 'Test Company' , Email = 'johntest@xyz.com');
        insert lead;
        
        RestRequest req = new RestRequest();
        req.addHeader('Content-Type', 'application/json');
        req.requestURI = '/services/apexrest/leadconvert/' + lead.Id;
        req.httpMethod = 'POST';
        RestContext.request = req;
        
        RestLeadConvert.doPost();
        System.assertEquals(RestLeadConvert.StatusCodeOK, RestContext.response.statusCode,'The Response is 200');
        
        
        RestRequest req2 = new RestRequest();
        req2.addHeader('Content-Type', 'application/json');
        req2.requestURI = '/services/apexrest/leadconvert/invalidLeadId';
        req2.httpMethod = 'POST';
        RestContext.request = req2;
        
        RestLeadConvert.doPost();
        System.assertEquals(RestLeadConvert.StatusCodeBadRequest, RestContext.response.statusCode,'The Response is 400');
        
        
    }
}